Study Guide for PART II of the Spring 14 MAT187 Final Exam. NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 5 multiple choice questions. You will be provided with two sheets of scratch paper which must both be turned in with you exam regardless of whether or not you use the scratch paper. You may NOT use your own scratch paper. This portion of the exam covers the Trigonometry portion of this course (4.1 4.1). You should focus on your notes and homework from those sections and Tests #5 and #6. Of course some of you may benefit from utilizing the supplemental reading and videos on the class Help page from these sections too! E SURE THAT YOU HAVE LOOKED AT, THOUGHT AOUT AND TRIED THE SUGGESTED PROLEMS ON THIS REVIEW GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!! Here is a brief overview of what you should be able to do! 1. e able to convert a degree measure into a radian measure (where will be part of your answer). Try 5 5 radians 18 4. e able to evaluate a trig function at a particular degree measure. Example: cos 1 1 just use the unit circle! 4. e able to utilize you unit circle to find the value of trig expressions like cos sin 4 1 cos sin 1 again, just use the unit circle! 4. Know which trig functions are positive and which ones are negative in each quadrant! S A 55 T C 5. Given a standard angle on the unit circle, be able to identify the coordinates. Having a solid understanding of the unit circle will help you on MANY of these problems! 6. e able to find the value of an inverse trig function expression. Example: and arcsin problems from your notes and homework too. arctan angle is arctan. e sure to practice arcos draw the partial unit circle from to and go find the only place where the tangent of that 6, note the answer is NOT 11 6 6 7. e able to find the value of composed trig functions and inverse trig functions. Example: Find others in your notes and homework to practice! 5 1 sin arccos sin 6 sin arccos.
8. e able to evaluate various trig functions at certain degree measures. Example: cos, cos1,sin6, sin6 Answers are,,, respectively 9. e able to calculate the length of an arc when given the radius and angle (in radians). See section 4.4 S r, in radians! 1. Can you solve x cos sin x? If x is an acute angle bigger than the relationship between the sine and cosine values of the two acute angles in a triangle are!? The key to this is remembering what note that if x is acute bigger than degrees then both x- and x are acute. If the cosine of an acute angle = the sine of another acute angle the two angles must add up to 9 (think about a right triangle and the two acute angles inside). x x 9 x 55 11. e able to simplify a basic trigonometric expression. Example: cos cot sin cos sin cos csc 1 sin 1 sin 1. Can you simplify 1 sin x1 sin x? 1sin x 1sin x 1sin x sin x sin x 1sin x cos x 1. What is sin6 cos4 cos6 sin 4 equivalent to? cot csc cos1,sin1,cos,sin. sin 6 cos4 cos6 sin 4 sin 6 4 sin Make sure that you know the trig identities that I told you were important! You should recognize THIS example as an example of the sine of the difference of two angles. e sure to think about what this problem would look like if it led to the sine of the sum of two angles or the cosine of the sum or difference of two angles. 5 sin A and cos and A and are both acute angles what does cos A? 14. If 5 1 What about sin A? To solve this problem draw two right triangles, one for angle A and one for angle. On the triangle with angle A in it you can solve for the missing side and get x = 4 and the one with angle in it and get y = 1. Then just remember the correct sum and difference formulas. 4 5 1 6 16 cos A cos Acos sin Asin 5 1 5 1 65 65 65 5 4 1 15 48 sin A sin Acos cos Asin 5 1 5 1 65 65 65 15. e able to simplify trigonometric expressions involving double angles. (go look some up!) e sure that you are familiar with the double angle identities for sine and cosine (# and #1). e sure that you know all three versions of #1!
16. e able to solve basic trig equations in restricted domains. Example: Solve tan x in 18, 6 tan x tan x x Use the unit circle on the given interval to look up the only place where y/x is C 17. Given a modified Sine equation be able to figure out the amplitude. y Asinx D Amplitude = A 18. Given a modified Sine equation be able to figure out the period. Period = 19. Given a modified Sine equation be able to figure out the horizontal shift. Horizontal shift = C. C Note if is negative (after you have rewritten the equation in "standard form") C then the horizontal shift is to the left.. Given the graph of a modified Sine function be able to select the corresponding equation. 1. Given a modified Sine equation be able to pick out the correct graph.. Make sure that you know the domains of Sine, Cosine and Tangent. Domain of both the sine and cosine function is, k Domain of the tangent function is all real numbers except where k is any odd integer. Remember that the tangent function has vertical asymptotes at any location where the cosine function =. Given two sides of a triangle and the included angle, be able to estimate the area. 1 1 1 Area absin C bcsin A acsin 4. Given two angles and one side of an angle, be able to find one of the remaining sides. Law of sines (unless it is a SAS in which case use law of cosines). b a b sin a sin sin A sin A 4 5. If cos A. What is the value of sin A? Make sure that YOU know all three versions of identity #1! 5 4 4 cos A 1 sin A 1 sin A 1 sin A 5 5 5 4 49 49 sin A sin A sin A 5 5 5 5 sin A 49 7 5 5
6. e able to use the Law of Sines to solve problems like. In triangle AC, 6 c 8 and sin C Find the length of side b 4 1 b c c 8 b sin sin 6 16 8 sin sin C sin C 1 4 7. e able to solve a SSS triangle for the largest angle. Since you are not allowed a calculator then obviously the side lengths must be chosen in such a way to yield an angle that you can obtain WITHOUT the use of a calculator! 8. e able to solve a SAS triangle for the missing side. You should be able to estimate the answer without a calculator given your knowledge of the few basic decimals I told you to know. 1.7.866 1.414.77 9. e able to find an exact value for a trig expression involving lesser used trig functions. Example: csc6 cot csc 45 1 cos 1 1 1 csc 6 cot csc 45 sin 6 sin sin 45 1 or. e able to compute compositions of regular trig functions with inverse trig functions. Like #7 UT different trig and inverse trig functions AND you will need to draw a triangle rather than use the unit circle! 1. Given an angle in standard position and the coordinates of a point on the terminal side be able to obtain the value of any trig function for that angle.. Can you find the exact value of sin 15 tan15? sin 15 tan15 1 1 or. If cos is in quadrant IV, find the value of sin 4 and. Draw a right triangle in quadrant IV with in standard position. The reference angle for will be one of the angles in your right triangle. Label the side adjacent to (the side on the positive x-axis) and label the hypotenuse 4. Use the Pythagorean theorem to find the y value and obtain 7. Then use your trig identity for sin. 7 7 sin sin cos 4 4 8
4. Given a modified cosine equation, be able to find the period. 5. What is cos7x cosx sin 7x sin x equivalent to? e sure you know the sum and difference identities for both sine and cosine. cos 7x cosx sin 7x sin x cos 7x x cos 1x 6. e able to simplify a trig expression involving cos Remember cos cos sin cos 1 1 sin 7. Make sure that you know identity #9 and the several other versions of it that we have used in this course. cos sin 1 cos 1sin sin 1 cos 8. Make sure that you understand the relationship between a trigonometric function and its inverse. 1 If cos A x then cos x A 9. e able to find a reference angle for a given degree measure. 4. Can you find an angle x in the third quadrant where x cot tan x? What about in the other quadrants? Go look at your unit circle! Remember that cotangent is x/y and tangent is y/x. Can you find an angle whose tangent matches the cotangent of an angle degrees less? (Answer 41. Can you find an angle coterminal to another angle? x 15 ). What about the other quadrants? 1 1 1 4. e sure to know for what values sin,sin,cos,cos, tan and tan are defined. i.e. What are the domains of each? sine and cosine both have domain,. tangent has domain all real numbers except 1 1 sin and cos both have domain 1,1 1 tan has domain, k where k is any odd integer. 4. e able to find a solution to a trig equation involving sine and cosine. 1 1 44. Can you find cot 1? sec? 1 1 1 1 cot 1 tan tan (1) 1 4 1 1 1 sec cos 45. e able to solve a very basic sine or cosine equation in the interval,6 46. Can you simplify basic expressions like cot x cos x? cos x cot x sin x cos x 1 1 csc x cos x cos x sin x cos x sin x 1
47. e sure that you know the word definitions for each of the six regular trig functions. i.e. opposite sin hypotenuse 48. e sure that you know the range of each of arcsin x, arc cos x and arctan x,,, respectively! 49. Know when you should use each of the following to find the missing side of a triangle.. Pythagorean theorem, Law of Sines, Law of Cosines, Similar triangles. 5. e able to calculate the area of a triangle when you are given two sides and the included angle. 1 1 1 Area absin C bcsin A acsin