SAT II Math Level 2 Study Guide pdf. SAT Subject Math Level 1 Practice Test from Official Study Guide pdf download · ARCO: SAT II Math 10th Edition from. Book Descriptions Introduction. The purpose of this book is to help you prepare for the SAT Level 2. Mathematics Subject Test. This book can be used as a. The newly updated tenth edition of Barron s SAT Subject Test Math 2 offers students all A diagnostic test with explained answers to help students identify their.
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I would also like to thank Barron's editor Pat Hunter for guiding me through the . OVERVIEW OF THE LEVEL 2 SUBJECT TEST The SAT Mathematics Level 2. Read Barron's SAT Subject Test PDF - Math Level 2, 12th Edition by Richard Ku M.A. Barron's Educational Series | This manual opens with a d. Barron's SAT Subject Test Math Level 2, 10th edition Barron's SAT Subject Test in Chemistry will be very helpful. five questions on equation balancing.
The reflection is vertical. One sign change indicates there will be exactly one negative zero of P x. B Factor and reduce. Suppose you have 5 shirts, 4 pairs of pants, and 9 ties. Use the law of sines: Moreover, the scale factor a must be factored out of a translation.
TIP Leave your cell phone at home, in your locker, or in your car! Your raw score is the number of correct answers minus one-fourth of the number of incorrect answers, rounded to the nearest whole number.
For example, if you get 30 correct answers, 15 incorrect answers, and leave 5 blank, your raw score would be , rounded to the nearest whole number. Raw scores are transformed into scaled scores between and The formula for this transformation changes slightly from year to year to reflect varying test difficulty. In recent years, a raw score of 44 was high enough to transform to a scaled score of Each point less in the raw score resulted in approximately 10 points less in the scaled score.
For a raw score of 44 or more, the approximate scaled score is For raw scores of 44 or less, the following formula can be used to get an approximate scaled score on the Diagnostic Test and each model test: The self-evaluation page for the Diagnostic Test and each model test includes spaces for you to calculate your raw score and scaled score. Although most testing centers have wall clocks, you would be wise to have a watch on your desk. Since there are 50 items on a one-hour test, you have a little over a minute per item.
Typically, test items are easier near the beginning of a test, and they get progressively more difficult. Work the problems you are confident of first, and then return later to the ones that are difficult for you.
Answer the question asked, not the one you may have expected. For example, you may have to solve an equation to answer the question, but the solution itself may not be the answer. Since you may skip questions that are difficult, be sure to mark the correct number on your answer sheet.
If you change an answer, erase cleanly and leave no stray marks. Mark only one answer; an item will be graded as incorrect if more than one answer choice is marked. It is usually not a good idea to change an answer on the basis of a hunch or whim. Even though calculators simplify the computational process, you may save time by identifying a pattern that leads to a shortcut.
If a problem contains only variable quantities, it is sometimes helpful to substitute numbers to understand the relationships implied in the problem. An appropriate percentage for you may differ from this, depending on your experience with calculators.
Even if you learned the material in a highly calculator-active environment, you may discover that a problem can be done more efficiently without a calculator than with one. If the answer choices are in decimal form, the problem is likely to require the use of a calculator. This should be taken under test conditions: Correct the test and identify areas of weakness using the cross-references to the Part 2 review.
Use the review to strengthen your understanding of the concepts involved. Ideally, you would start preparing for the test two to three months in advance. Each week, you would be able to take one sample test, following the same procedure as for the Diagnostic Test. Depending on how well you do, it might take you anywhere between 15 minutes and an hour to complete the work after you take the test.
Obviously, if you have less time to prepare, you would have to intensify your efforts to complete the six sample tests, or do fewer of them. The best way to use Part 2 of this book is as reference material. You should look through this material quickly before you take the sample tests, just to get an idea of the range of topics covered and the level of detail. Please adjust your device accordingly. Since this is an e-Book, record all answers and self-evaluations separately.
The diagnostic test is designed to help you pinpoint your weaknesses and target areas for improvement. The answer explanations that follow the test are keyed to sections of the book. To make the best use of this diagnostic test, set aside between 1 and 2 hours so you will be able to do the whole test at one sitting. Tear out the preceding answer sheet and indicate your answers in the appropriate spaces.
Do the problems as if this were a regular testing session. When finished, check your answers against the Answer Key at the end of the test.
For those that you got wrong, note the sections containing the material that you must review. If you do not fully understand how to get a correct answer, you should review those sections also.
The Diagnostic Test questions contain a hyperlink to their Answer Explanations. Simply click on the question numbers to move back and forth between questions and answers. Finally, fill out the self-evaluation on a separate sheet of paper in order to pinpoint the topics that gave you the most difficulty. Decide which answer choice is best. If the exact numerical value is not one of the answer choices, select the closest approximation.
Fill in the oval on the answer sheet that corresponds to your choice. Figures are drawn as accurately as possible to provide useful information for solving the problem, except when it is stated in a particular problem that the figure is not drawn to scale. TIP For the Diagnostic Test, practice exercises, and sample tests, an asterisk in the Answers and Explanations section indicates that a graphing calculator is necessary. Reference Information. The following formulas are provided for your information.
Volume of a right circular cone with radius r and height h: Lateral area of a right circular cone if the base has circumference C and slant height is l: Volume of a sphere of radius r: Surface area of a sphere of radius r: A linear function, f, has a slope of —2.
Find q. Which of the following is not an even function? What is the radius of a sphere, with center at the origin, that passes through point 2,3,4? If a point x,y is in the second quadrant, which of the following must be true? The average of your first three test grades is What grade must you get on your fourth and final test to make your average 80? A 0 B 7 C 9 D 11 E an infinite number A linear function has an x-intercept of and a y-intercept of. The graph of the function has a slope of A —1.
Given the set of data 1, 1, 2, 2, 2, 3, 3, 4, which one of the following statements is true? If , what is the value of?
Find all values of x that make. The statistics below provide a summary of IQ scores of children. The height of a cone is equal to the radius of its base. The radius of a sphere is equal to the radius of the base of the cone. Which of the following lines are asymptotes of the graph of? What is the probability of getting at least three heads when flipping four coins? In the figure above, S is the set of all points in the shaded region. Which of the following represents the set consisting of all points 2x,y , where x,y is a point in S?
If a square prism is inscribed in a right circular cylinder of radius 3 and height 6, the volume inside the cylinder but outside the prism is A 2. The fifth term of an arithmetic sequence is 26, and the eighth term is What is the first term? A A only 1 B only 0 C all real numbers D all real numbers except 0 E no real numbers For what value s of k i s F a continuous function?
If , what is the value of f —1 15? Which of the following could be the equation of one cycle of the graph in the figure above? Observers at locations due north and due south of a rocket launchpad sight a rocket at a height of 10 kilometers.
How far apart are the two observers if their angles of elevation to the rocket are The length of the base is 10 centimeters. How many centimeters are in the perimeter? A left 2 units and up k units B right 2 units and up k— 4 units C left 2 units and up k— 4 units D right 2 units and down k— 4 units E left 2 units and down k— 4 units A certain component of an electronic device has a probability of 0.
If there are 6 such components in a circuit, what is the probability that at least one fails? The number in brackets after each explanation indicates the appropriate section in the Review of Major Topics Part 2. If a problem can be solved using algebraic techniques alone, [algebra] appears after the explanation, and no reference is given for that problem in the Self-Evaluation Chart at the end of the test.
An asterisk appears next to those solutions for which a graphing calculator is necessary. Graph each answer choice to see that Choice D is not symmetric about the y-axis.
An alternative solution is to use the fact that sin x sin —x , from which you deduce the correct answer choice. Odd x even is always an odd function.
E A point in the second quadrant has a negative x-coordinate and a positive y-coordinate. The correct answer is E. C f a means to replace x in the formula with an a. If x represents your final test grade, the average of the four test grades is , which is to be equal to A Add the two equations: You can count 11 integers between —3 and 7 if you include both endpoints. D indicates the need for the positive square root of x2.
Therefore, if and if. This is just the definition of absolute value, and so is the only answer for all values of x. An alternative solution is to use the facts that has vertical asymptotes when the denominator is zero, i. Divide the numerator and denominator of the expression by x2 and observe that the expression approaches as. Use the x-intercept to get and the y-intercept to get. Therefore, and. B Substituting the points into the equation gives , and.
Divide through by 6y so that will be on one side of the equals sign. This gives. The 2 by 2 matrix on the right side of the equation has the determinant x2— An alternative solution is to recognize that the graph of f is a parabola that is symmetric about the y-axis and opens up. The sine of a first quadrant angle is positive, but the sine of a third quadrant angle is negative. Since f a f b , it follows that a b. C The slope of the given line is —3. Therefore, the slope of a perpendicular line is the negative reciprocal, or.
C Fifty children is half of children, and half of the data points lie between the first and third quartiles. An alternative solution is to recall that cos x is even, so its reciprocal is also even. A From the figure. Therefore, there are distinguishable ways the letters can be arranged. At least 3 heads means 3 or 4 heads. C Since the y values remain the same but the x values are doubled, the circle is stretched along the x-axis. Therefore, the desired volume is 54 — , which using your calculator is approximately The length of the major axis is [2.
D There are three constant differences between the fifth and eighth terms.
The fifth term, 26, is four constant differences 20 more than the first term. The larger angle is the supplement, or Since k is in the denominator, it cannot equal 0.
Enter values of x progressively closer to 1 e. C , and. C f —1 15 is the value of x that makes equal to Set , divide both sides by 3 to get. An alternative solution is to deduce facts about the graphs from the equations. All three equations indicate graphs that have period. The graph of equation I is a normal sine curve. The graph of equation II is a cosine curve with a phase shift right of , one-fourth of the period. Therefore, it fits a normal sine curve. The graph of equation III is a sine curve that has a phase shift left of , one-half the period, and reflected through the x-axis.
This also fits a normal sine curve. An alternative solution is to distribute and transform the equation to read: For , there are three solutions: C The problem information is illustrated in the figure below.
Points A and B represent the two observers. Point C is the base of the altitude from the rocket to the ground. We know that and. Therefore, [1. D Drop the altitude from the vertex to the base. The altitude bisects both the vertex angle and the base, cutting the triangle into two congruent right triangles. C The given graph looks like the left half of a parabola with vertex 4,—3 using the values given in the answer choices as guides that opens up. Therefore, b0.
From the figure,. Since the probability of one component succeeding is 1 minus 0. A sound understanding of these concepts certainly will improve your score. The techniques discussed may help you save time solving some of the problems without a calculator at all. For problems requiring computational power, techniques are described that will help you use your calculator in the most efficient manner.
Your classroom experience will guide your decisions about how best to use a graphing calculator. If you have been through a secondary mathematics program that attached equal importance to graphical, tabular, and algebraic presentations, then you probably will rely on your graphing calculator as your primary tool to help you find solutions.
However, if you went through a more traditional mathematics program, where algebra and algebraic techniques were stressed, it may be more natural for you to use a graphing calculator only after considering other approaches. Functions are usually specified by equations such as. In this equation x represents an input number while y represents the unique corresponding output number. Functions can also have names: Taken as a group, the input numbers are called the domain of the function, while the output numbers are called the range.
Unless otherwise specified, the domain of a function is all real numbers for which the equation produces outputs that are real numbers. In this case, the range is the set of all non-negative numbers. The domain of a function can also be established as part of the definition of a function. Unless a domain is explicitly stated, the domain is assumed to be all real values that produce real numbers as outputs.
A function with a small finite domain can be described by a set of ordered pairs instead of an equation. The first number in the pair is from the domain and the second is the corresponding range value.
The domain of this function consists of 0, 1, 2, and 3, while the range consists of 2, 1, 3, and 8. Functions like this are typically used to illustrate certain properties of functions and are discussed later.
A function is actually a special type of relation. A relation describes the association between two variables. All ordered pairs x, y that satisfy the equations are in the relation. In this case, these pairs form the circle of radius 2 centered at the origin. TIP Typically, a value of x that must be excluded from the domain of a function makes the denominator zero or makes the value of an expression under a radical less than zero.
Other than circles, relations that are not functions include ellipses, hyperbolas, and parabolas that open right or left, instead of up or down—in other words, the conic sections discussed in Section 2. Like functions, relations can also be defined using specific ordered pairs. A 3,2 B 4,2 C 2,3 D 7,1 E none of the above 2. What value s must be excluded from the domain of? All answers to exercises appear at the end of each section. Resist the urge to peek before trying the problems on your own.
The inverse of a function is not necessarily a function. To verify this, proceed as follows: Find the inverse. To verify this, check f f —1 and f —1 f term by term. In this case f —1 is not a function. If the point with coordinates a,b belongs to a function f, then the point with coordinates b,a belongs to the inverse of f. Algebraically, the equation of an inverse of a function can be found by replacing f x by y; interchanging x and y; and solving the resulting equation for y.
In order to find f —1, interchange x and y and solve for y: Thus, 6. The inverse of any function f can always be made a function by limiting the domain of f. In Example 6 the domain of f could be limited to all nonnegative numbers or all nonpositive numbers. In this way f —1 would become either or , both of which are functions. Then switch x and y: Solve for y: Find the range of. First replace f x by y, and interchange x and y to get.
Then solve for y: This could also be determined by observing that in the original function can never be zero. Which of the following could represent the equation of the inverse of the graph in the figure? The graph of an even relation or function is symmetric with respect to the y axis. TIP An even relation is symmetric about the y-axis. A relation is said to be odd if —x,—y is in the relation whenever x,y is. If the relation is defined by an equation, it is odd if —x,—y satisfies the equation whenever x,y does.
The graph of an odd relation or function is symmetric with respect to the origin. TIP An odd relation is symmetric about the origin. TIP Relations can be either odd, even, or neither.
They can also be both odd and even! The sum of even functions is even. The sum of odd functions is odd. The product of an even function and an odd function is odd.
The product of two even functions or two odd functions is even. Which of the following relations are even? Which of the following relations are odd? Which of the following relations are both odd and even? Which of the following functions is neither odd nor even? A Either 3,2 or 3,1 , which is not an answer choice, must be removed so that 3 will be paired with only one number. E For each value of x there is only one value for y in each case.
Therefore, f, g, and h are all functions. C Since division by zero is forbidden, x cannot equal 2. Combining Functions 1. C To get from f x to f g x , x2 must become 4x2. D g x cannot equal 0. A By definition. Therefore, the domain of the original function must lose either 1 or 5. The only possibilities are Choices D and E. Choice D can be excluded because since the x-intercept of f x is greater than —1, the y- intercept of its inverse must be greater than —1. Odd and Even Functions 1.
D Use the appropriate test for determining whether a relation is even. D Use the appropriate test for determining whether a relation is odd. C A is even, B is odd, D is even, and E is odd.
The graph is always a straight line. The slope of the line is represented by m and is defined to be the ratio of , where x1, y1 and x2, y2 are any two points on the line. The y-intercept is b the point where the graph crosses the y-axis. If you solve the general equation of a line, you will find that the slope is and the y-intercept is.
You can always quickly write an equation of a line when given its slope and a point on it by using the point-slope form: If you are given two points on a line, you must first find the slope using the two points.
Then use either point and this slope to write the equation. Once you have the equation in point-slope form, you can always solve for y to get the slope-intercept form if necessary. Write an equation of the line containing 6,—5 and having slope. In point-slope form, the equation is. Write an equation of the line containing 1,—3 and —4,—2. First find the slope. Then use the point 1,—3 and this slope to write the point-slope equation.
Parallel lines have the same slope.
The slopes of two perpendicular lines are negative reciprocals of one another. These lines are parallel because the slope of each line is 2, and the y-intercepts are different. The equation of line l1 is , and the equation of line l2 is. These lines are perpendicular because the slope of l2, , is the negative reciprocal of the slope of l1, You can use these facts to write an equation of a line that is parallel or perpendicular to a given line and that contains a given point.
The slope of the given line is. The point-slope equation of the line containing 1,7 is therefore. The slope of the given line is 4, so the slope of a line perpendicular to it is. The desired equation is. The y-intercept of the line through the two points whose coordinates are 5,—2 and 1,3 is A B C D 7 E 17 5. The length of the segment joining the points with coordinates —2,4 and 3,—5 is A 2.
The graph is always a parabola. The x-coordinate of the vertex of the parabola is equal to , and the axis of symmetry is the vertical line whose equation is. To find the minimum or maximum value of the function, substitute for x to determine y. Thus, in general the coordinates of the vertex are and the minimum or maximum value of the function is.
Unless specifically limited, the domain of a quadratic function is all real numbers, and the range is all values of y greater than or equal to the minimum value or all values of y less than or equal to the maximum value of the function. The examples below provide algebraic underpinnings of how the orientation, vertex, axis of symmetry, and zeros are determined. You can, of course, use a graphing calculator to sketch a parabola and find its vertex and x-intercepts.
Does the quadratic function have a minimum or maximum value? If so, what is it? The equation of the axis of symmetry is and the y-coordinate of the vertex is The vertex is, therefore, at.
The minimum value is. The solutions are , the general quadratic formula. TIP Most numerical answer choices on the Math Level 2 test are in the form of numerical approximations. Simplified radical answer choices are rarely given. The left side does not factor. These solutions are most readily obtained by using the polynomial solver on your graphing calculator.
Note that the sum of the two zeros, equals , and their product equals. This information can be used to check whether the correct zeros have been found. In Example 2, the sum and product of the zeros can be determined by inspection from the equations. TIP The sum of the zeros is and the product of the zeros is. At times it is necessary to determine only the nature of the roots of a quadratic equation, not the roots themselves.
Because b2 — 4ac of the general quadratic formula is under the radical, its sign determines whether the roots are real or imaginary. The quantity b2 — 4a c is called the discriminant of a quadratic equation.
TIP If the zeros are complex, the parabola does not cross the x-axis. A parabola with a vertical axis has its vertex at the origin and passes through point 7,7. The length of the segment joining these points is A 14 B 13 C 12 D 8. Here are five facts about the graphs of polynomial functions: They are always continuous curves. The graph can be drawn without removing the pencil from the paper.
If the largest exponent is an odd number, the ends of the graph leave the coordinate system at opposite ends. Facts 2 and 3 describe the end behavior of a polynomial. If all the exponents are even numbers, the polynomial is an even function and therefore symmetric about the y-axis.
If all the exponents are odd numbers and there is no constant term, the polynomial is an odd function and therefore symmetric about the origin of the coordinate system. The zeros of polynomials with real coefficients can be real or imaginary numbers, but imaginary zeros must occur in pairs. For example, if the degree of a polynomial is 6, there are 6 real zeros and no imaginary zeros; 4 real zeros and 2 imaginary ones; 2 real zeros and 4 imaginary ones, or 6 imaginary zeros.
Moreover, real zeros can occur more than once. If a real zero occurs n times, it is said to have multiplicity n. If a zero of a polynomial has odd multiplicity, its graph crosses the x-axis. The multiplicity of a real zero is counted toward the total number of zeros. There are 5 facts that are useful when analyzing polynomial functions. Remainder theorem—If a polynomial P x is divided by x — r where r is any constant , then the remainder is P r.
The remainder is —6 when you divide P x by x — 3. Factor theorem —r is a zero of the polynomial P x if and only if x — r is a divisor of P x. Call this polynomial P x and evaluate P 99 using your graphing calculator. Rational zero root theorem—If is a rational zero reduced to lowest terms of a polynomial P x with integral coefficients, then p is a factor of a0 the constant term and q is a factor of an the leading coefficient.
If P x is a polynomial with real coefficients, then complex zeros occur as conjugate pairs. The number of negative real zeros of P x either is equal to the number of changes of the sign between the terms of P —x or is less than that number by an even integer.
Three sign changes indicate there will be either one or three positive zeros of P x. One sign change indicates there will be exactly one negative zero of P x. Which of the following is an odd function?
You need to find the x values of points on the graph that lie below the x-axis. First find the zeros: The slope is. The slope of a perpendicular line is. C The slope of the line is , so the point-slope equation is. Solve for y to get. The y-intercept of the line is.
C The slope of the segment is. Therefore, the slope of a perpendicular line is —3. The midpoint of the segment is. Therefore, the point-slope equation is. Therefore, the slope of a parallel line.
C The slope of the first line is , and the slope of the second line is. To be perpendicular,. Quadratic Functions 1. Hence the vertex is the point —1,—7. C Find the vertex: B The x coordinate of the vertex is. Thus, the equation of the axis of symmetry is. The zeros are and —2.
D Sum of. Substitute 7,7 for x and y to find. Higher-Degree Polynomial Functions 1. A Since the degree of the polynomial is an even number, both ends of the graph go off in the same direction. Since P x increases without bound as x increases, P x also increases without bound as x decreases. C Since the exponents are all odd, and there is no constant term, III is the only odd function.
E Rational roots have the form , where p is a factor of 12 and q is a factor of 2. The total is D Substitute 3 for x set equal to zero and solve for K. Inequalities 1. Numbers between these satisfy the original inequality. C Graph the function, and determine that the three zeros are —1. The parts of the graph that are above the x-axis have x-coordinates between —1. When an angle is placed so that its vertex is at the origin, its initial side is along the positive x-axis, and its terminal side is anywhere on the coordinate system, it is said to be in standard position.
The angle is given a positive value if it is measured in a counterclockwise direction from the initial side to the terminal side, and a negative value if it is measured in a clockwise direction. Let P x,y be any point on the terminal side of the angle, and let r represent the distance between O and P.
The six trigonometric functions are defined to be: From these definitions it follows that: The distance O P is always positive, and the x and y coordinates of P are positive or negative depending on which quadrant the terminal side of lies in.
The signs of the trigonometric functions are indicated in the following table. TIP All trig functions are positive in quadrant I. Sine and only sine is positive in quadrant II. Tangent and only tangent is positive in quadrant III. Cosine and only cosine is positive in quadrant IV. Just remember: All Students Take Calculus. Each angle whose terminal side lies in quadrant II, III, or IV has associated with it an angle called its reference angle , which is formed by the x-axis and the terminal side.
The sign is determined by the quadrant in which the terminal side lies. Sine and cosine, tangent and cotangent, and secant and cosecant are cofunction pairs. Cofunctions of complementary angles are equal. Since these cofunctions are equal, the angles must be complementary. A radian is one radius length. The circle shown in the figure below has radius r. In each of the following, convert the degrees to radians or the radians to degrees.
If no unit of measurement is indicated, radians are assumed. The length of the arc, s, is equal to r , and the area of the sector, AOB, is equal to. Find the area of the sector and the length of the arc subtended by a central angle of radians in a circle whose radius is 6 inches.
An angle of 30 radians is equal to how many degrees? If a circle has a circumference of 16 inches, the area of a sector with a central angle of 4. Find s. How long is the pendulum? The ratios of the sides of the two special triangles are shown in the figure below. You can now use the definitions of the trig functions to find the trig values: Values can be checked by comparing the decimal approximation the calculator provides for the trig function with the decimal approximation obtained by entering the exact value in a calculator.
The terminal sides of these angles are the x- and y-axes. Such an analysis can determine the amplitude, maximum, minimum, period, or phase shift of a trig function, or solve a trig equation or inequality. The examples and exercises in this and the next two sections show how a variety of trig problems can be solved without using a graphing calculator. They also explain how to solve trig equations and inequalities and how to analyze inverse trig functions.
The smallest positive value of p for which this property holds is called the period of the function. The sine, cosine, secant, and cosecant have periods of 2 , and the tangent and cotangent have periods of.
The domain and range of each of the six trigonometric functions are summarized in the table. They can be translated slid horizontally or vertically or dilated stretched or shrunk horizontally or vertically.
The parameters A and D accomplish vertical translation and dilation, while B and C accomplish horizontal translation and dilation. When working with trig functions, the vertical dilation results in the amplitude, whose value is A. The horizontal translation is and is called the phase shift, and the horizontal dilation of trig functions is measured as the period, which is the period of the parent trig function divided by B.
Finally, D is the amount of vertical translation. TIP The frequency of a trig function is the reciprocal of its period. Graphs of the parent trig functions follow. TIP sin x is an odd function. TIP Note: Determine the amplitude, period, and phase shift of and sketch at least one period of the graph. The graph can continue to the right and to the left for as many periods as desired.
Since the coefficient of the sine is negative, the graph starts down as x increases from —3, instead of up as a normal sine graph does. What are the coordinates of point P? As increases from , the value of A increases, and then decreases B decreases, and then increases C decreases throughout D increases throughout E decreases, increases, and then decreases again 4. For what value of P is the period of the function Px equal to? If , what is the maximum value of the function?
Using radian measure: From the figure below, you can see that sin. Therefore, sin. You may be expected to solve trigonometric equations on the Math Level 2 Subject Test by using your graphing calculator and getting answers that are decimal approximations.
To solve any equation, enter each side of the equation into a function Yn , graph both functions, and find the point s of intersection on the indicated domain by choosing an appropriate window. Solutions x-coordinates of intersection points are 1. Enter each side of the inequality into a function, graph both, and find the values of x where the graph of cos x lies beneath the graph of sin 2x: If sin and cos , find the value of sin 2x.
If cos , find cos 2x. Since all trig functions are periodic, graphs of their inverses are not graphs of functions. The domain of a trig function needs to be limited to one period so that range values are achieved exactly once.
The inverse of the restricted sine function is sin—1; the inverse of the restricted cosine function is cos—1, and so forth. The inverse trig functions are used to represent angles with known trig values. If you know that the tangent of an angle is , but you do not know the degree measure or radian measure of the angle, tan — 1 is an expression that represents the angle between whose tangent is. You can use your graphing calculator to find the degree or radian measure of an inverse trig value.
Evaluate the radian measure of tan—1. Enter 2nd tan with your calculator in radian mode to get 0. Evaluate the degree measure of sin—1 0.
Enter 2nd sin. Evaluate the degree measure of sec—1 3. Therefore, enter 2nd cos with your calculator in degree mode to get However, because of the range restriction on inverse trig functions, trig—1 trig x need not equal x.
Evaluate cos cos—1 0. Evaluate sin. Then cos and x is in the first quadrant. See the figure below. Find the number of degrees in. A —45 B — Find the number of radians in cos—1 —0. Find the number of radians in cot—1 —5. A — Which of the following is are true? Depending on which of the sides and angles of the triangle are supplied, the following formulas can be used to find missing parts of a triangle.
Law of Cosines: Find the number of degrees in the largest angle of a triangle whose sides are 3, 5, and 7. The largest angle is opposite the longest side. Use the Law of Cosines: Therefore, cos. Use the law of sines: Ambiguous Cases If the lengths of two sides of a triangle and the angle opposite one of those sides are given, it is possible that two triangles, one triangle, or no triangle can be constructed with the data.
This is called the ambiguous case. If the lengths of sides a and b and the value of A are given, the length of side b determines the number of triangles that can be constructed. Case 1: Let the length of the altitude from C to the base line be h. Case 2: If a compass is opened the length of side a and a circle is drawn with center a t C, the circle will cut the baseline at two points, B1 and B2.
Which of the following must be true? The angles of a triangle are in a ratio of 8: The ratio of the longest side of the triangle to the next longest side is A B 8: The sides of a triangle are in a ratio of 4: What are all values of side a in the figure below such that two triangles can be constructed? Given the following data, which can form two triangles?
Cosine in quadrant IV is positive. D See corresponding figure. Therefore, tan. A Cofunctions of complementary angles are equal.
A Angle is in quadrant II, and sin is positive. Angle is in quadrant IV, and sin is negative. Arcs and Angles 1. Special Angles 1. C Sketch an angle of radians in standard position, as shown in the figure below.
The cosine ratio is. In this case, one subtraction suffices. Graphs 1. C Period. Point P is of the way through the period. Amplitude is 1 because the coefficient of sin is 1. Therefore, point P is at. Graph translated unit up. Graph looks like a cosine graph reflected about x-axis and shifted up unit. C Graph the function and determine its maximum 2 and minimum — 2. Subtract and then divide by 2. C Graph the function using 0 for Xmin and for Xmax. Phase shift for a sine curve in the figure is —.
Observe that the first graph is beneath the second on [0,0. D Remember that the range of the sine function is [—1,1], so the second term ranges from 6 to —6.
E Set your calculator to degree mode, and enter 2nd sin—1. D Set your calculator to radian mode, and enter 2nd cos—1 —0. E The range of inverse cotangent functions consists of only positive numbers. A Since , I is true. Since the range of cos—1 is [0, ], III is not true because cos—1 can never be negative.
Triangles 1. D Law of Sines: E By the Law of Sines: Let c be the longest side and b the next longest. D Use the Law of Cosines. Let the sides be 4, 5, and 6. C Law of Cosines: Therefore, d Law of Sines: D Area. C Area. At this point in the solution you know there have to be two values for C. Therefore, the answer must be Choice C or E. Therefore, Choice E is not the answer, and so Choice C is the correct answer.
A In I the altitude and so 2 triangles. In III the altitude so no triangle. The basic exponential properties: For all positive real numbers x and y, and all real numbers a and b: The basic logarithmic properties: For all positive real numbers a, b, p, and q, and all real numbers x, where a 1 and b 1: The basic property that relates the exponential and logarithmic functions is: If the base is the number e, ln, the natural logarithm, is used instead of loge.
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