Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy I: Changing to sines and cosines csc cot 1. sec Strategy II: Pythagorean or Co-Function Identities. sin x cos x csc x cot x 3. sin tan cos cos Strategy III: Factoring the GCF 3 cot w cot w 4. cos w Strategy IV: Even-Odd Identities 5. cot( x) cot x 6. tan( ) csc 1 We can use Identities to simplify trigonometric expression, but there is also a visual explanation for why we can simplify. Before we get into proving identities, try this exploration to further your understanding 7. Use your graphing calculator, set in radian mode, to complete the following: 3 a. Graph the function y sin x cos xsin x. Note: you will need to enter the exponents using parenthesis around the trig functions, such as sin( x ) 3. b. Select Zoom 7: ZTrig for a good window. c. Explain what you see and sketch a graph. d. Write an Identity for the expression you graphed and what it equals. PROVE your identity!
For the remainder of this lesson, we will PROVE identities. Remember to indicate where you are starting and to show all steps that lead you to the other side. If you are stuck, think through Strategies I through IV and keep in mind what you are trying to prove!! Verify (Prove) each identity. 8. tan x = sin x + sin x tan x 9. sin x cos x sec x cos x 10. tan xcsc x sec x 1 11. sin x tan x cos x cot x cos x 1 cos x 1. cos = cot sin csc 13. tan( x) tan x 1
16. sec tansec tan cos 17. sec sin ytan ycos y sec y sec y 18. sec w tan w 1 cos x sin x 19. csc 3 3 cot w cot w cos w w Warm Up: Fundamental Identities Day Perform the operation without a calculator 0. 5 3 1. 5 3. 8 4 8 7 x x 7 3. 5 3 x 1 4 x 1 5
Pre Calculus Worksheet: Fundamental Identities Day Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy and those strategies before. Strategy V: Combining Fractions with LCD 1 cos 1.. sin sin sin x 1 cos x 1 cosx sinx Strategy VI: Factoring with Difference of Squares 3. sec x 1 1 secx Strategy VII: Factoring trinomials with box 4 4 4. sin x sin xcos x cos x In general we don t want to leave a trigonometric expression with a fraction in it. Sometimes, however, we have no choice (you may have noticed we left a fraction on example 10 of the notes). If we do have to leave a fraction in our expression, we want to make it a nice fraction. For trigonometry, this means we prefer the fraction only have one trig. function and we prefer any addition or subtractions to be left in the numerator. To make this occur, we multiply by a special form of 1 similar to what we did with division involving i or division of radicals For example, 3 4 i i 11 i i or 1 5 5. Let s try it with trigonometric expressions. i 5 5 5 5 Multiply by 1
Strategy VIII: Multiply by 1 ASK if you are stuck!!! 5. cos x 1 sinx 6. tan x sec x 1 Verify (Prove) each identity. Use extra paper if needed!! 7. sin 1 sin cos 8. sec x sin x sin x cos x cot x 4 4 9. cos 10. sin csc csc sin 13cos 4cos 14 cos u u u u sin u 1 cos
11. cos csc sin sinsec 3 cot 1. sin sin csc 1cos 1cos 13. 1 tan tan 14. 1 cot 1 1 cot sec w1 sec w1 w Warm Up Lesson 5.3: Tell whether each statement is True or False. Then, given an example to justify your answer. 1. x y x y. x y x y 3. x 3 3 x 4 4 4. log xylog x log y 5. sin 3060 sin 30sin 60
Pre Calculus Worksheet 5.3 Introduction 1. A refresher as to why the sum/difference rules don t work the way many people want them to: a) Find sin( 30+ 60 ), and then find sin( 30 ) sin( 60 ) +. Are they the same? b) Find cos( 10+ 60 ), and then find cos( 10 ) cos( 60 ) +. Are they the same? c) Find tan( 60-30 ), and then find tan( 60 ) tan( 30 ) -. Are they the same? Now let s get a little deeper into where the sum and difference identities come from Cosine of Differences: cosa b a) Angles a and b are drawn below on one unit circle. Assume a band label these angles on the left picture below. Let a b. Notice you could have drawn angle θ in the first quadrant and it is the same size as θ from part (a). Label θ in the unit circle on the right. B (cos b, sin b) C (cos θ, sin θ) A (cos a, sin a) D (1, 0) b) Draw the chord created by the points where angle θ intersects each unit circle. Their coordinates are given. c) Do you agree these two chords are =? Then, their lengths according to the distance formula must be the same. This is the foundation of our derivation. The rest of the derivation contains Identities and Algebra. AB CD cos acosb sin asin b cos 1 sin 0
Using the Cosine of a Difference. Using cosa b, let s find cosa b cos ab cos acos b sin asin b, we can find 3 more identities.. a) From Algebra, subtraction is defined as adding the opposite. Use this definition to rewrite cosa b the difference of two angles. as b) Now apply Odd Identities. cos ab cos acos b sin asin b to what you wrote in part (a). Then, simplify using Even- 3. Using cosa b, let s find sin a b. a) From Fundamental Identities Day 1, recall we can use the Co-Function Identity: sin cos to sin a b. Let a b. rewrite the sine of an angle as a cosine function. Apply this identity to b) Next, distribute your negative to write your expression as the cosine of the difference of two angles. You will need to regroup. c) Now apply sin cos again. cos ab cos acos b sin asin b to what you wrote in part (a). Then, simplify using 4. Using sin a b, let s find sin a b. a) Again, subtraction is defined as adding the opposite. Use this definition to rewrite sin a b of two angles. as the sum b) Now apply sin ab sin acos b cos asin b to what you wrote in part (a). Then, simplify using Even- Odd Identities.
Pre Calculus Worksheet 5.3 1. Write the expression as the sine, cosine or tangent of a single angle. Then, evaluate if possible. a) sin cos cos sin 5 7 5 7 b) tan19tan 41 1 tan19tan41 tan tan 3 c) 1 tan tan 3 d) cos 6cos94sin 6sin 94. Use a sum or difference identity to find the exact value for each function. a) cos75 b) sin 195 c) cos 1 d) 11 tan 1
3. Simplify the following expressions as much as possible: p p a) sin( x + ) = b) tan( ) 6 q + = 4 4. Prove the following identities a) sin( x + y) + sin( x- y) = sin xcos y b) ( ) ( ) cos x + y + cos x- y = cos x cos y c) tan( x + p) -tan( p- x) = tan x d) ( ) ( ) cos cos cos sin x + y x- y = x- y 5. Use the function shown to answer the following questions. a) Write a sine function that fits the graph. b) Write a cosine function that fits the graph. c) Use identities to PROVE your answers from part a and b are the same.
6. Write each trigonometric expression as an algebraic expression. a) sin( arcsin x arccos x) - + b) ( 1 - cos sin x- tan 1 x) In preparation for lesson 5.4 7. Prove the following identities. a) sin x = sin xcos x b) cos cos sin x = x- x
PreCalculus Worksheet 5.4 For questions 1 and, write as the function of one angle. Simplify, if possible, without using a calculator.. sin cos 6 6 1. 1 sin 15 For questions 3 5, suppose sin A = 3 5 and A is an angle in the first quadrant, find each value. 3. cos (A) 4. tan (A) 5. sin (A) For questions 6 8, if tan y = 5 1 and y is an angle in the third quadrant, find each value. 6. sin (y) 7. tan (y) 8. cos (y)
For questions 9 14, prove each identity. Use a separate sheet of paper. tan( A) 9. sin( A) = 10. sin ( x) = cot( x) sin ( x) 1 + tan ( A) sin( x) 11. cot( x) = 1. ( x) é ( x) ( x) ù 1 - cos( x) ë û sin cot + tan = 13. csc( x) sec( x) = csc( x) 14. cos( 4x) = 1-8sin ( x) cos ( x) 3 15. sin( 3u) = 3cos usin u- sin u 16. ( ) + = ( ) cos 3x cos x cos x cos x
Pre Calculus Worksheet: Solving Trigonometric Equations For questions 1-6, solve each equation on the interval [0, π]. 1. cosx 5 4. sin xtan x sin x 0 3. cos x sin x 1 4. sin x 5sin x 0 5. cos x cosx 0 6. cos x cosx 1 7. Solve for x on the domain [0, ) : 1 cos x 8. Solve for x on the domain (, ): 1 cos x 9. Explain the difference in your solutions for questions 7-8.
Find all solutions to each equation. 10. 4cosx 3 cosx 11. sin x sin x 3 1. sin x 3sin x 0 13. cos x 1 14. cos x 4 7cos x 15. sin x sin xcos x 0 16. 3sin x cos x 0 17. 3 sin x 1