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 cscθ cotθ 1. secθ Strategy II: Pythagorean or Co-Function Identities. ( sin x cos x) ( csc x cot x) + 3. sinθ tanθ cosθ + cos( π θ) Strategy III: Factoring the GCF 3 cot w+ cot w 4. cos w Strategy IV: Even-Odd Identities π 5. cot( x) cot ( x) 6. tan( θ) ( csc θ 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 x csc x 1 sec x sin tan cos cot cos = 1 cos x x x x x x = 11. ( )( ) 1. cos θ = cot θ sin θ cscθ 13. π tan( x) tan x = 1
16. ( secθ tan θ)( secθ + tan θ) cosθ = 17. secθ sin y+ tan y+ cos y sec y = sec y 18. sec cos w tan ( x) ( w) + sin = 1 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 θ 1.. sin θ sin θ sin x 1 cos x + 1 cos x sin x Strategy VI: Factoring with Difference of Squares 3. sec x 1 1 sec x Strategy VII: Factoring trinomials with box 4 4 4. sin x sin x cos 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 sin x 6. tan x sec x 1 Verify (Prove) each identity. Use extra paper if needed!! 7. sin θ 1= sin θ cos θ 8. sec x sin x sin x cos x = cot x 4 4 9. cos θ θ = + θ θ 10. sin θ csc csc sin 1 3cos 4 cos 1 4 cos u u u = u sin u 1 cos
11. cos θ csc θ + sin θ sinθsecθ 3 = cot θ 1. sin β sin β csc β = + 1 cos β 1+ cos β 13. 1+ tanθ tanθ = 14. 1 + cotθ 1 1 = cot sec w 1 sec w+ 1 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 ( xy) = log x log y 5. ( ) sin 30 + 60 = sin 30 + sin 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 sin3060, and then find sin30 sin60. Are they the same? b) Find cos1060, and then find cos10 cos60. Are they the same? c) Find tan6030, and then find tan60 tan30. Are they the same? Now let s get a little deeper into where the sum and difference identities come from Cosine of Differences: cos( a 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 θ = 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 a cosb) + ( sin a sin b) = ( cosθ 1) + ( sinθ 0)
Using the Cosine of a Difference ( ). Using cos ( a b), let s find cos ( a b) cos a b = 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 cos ( a b) the difference of two angles. + as b) Now apply ( ) Odd Identities. cos a b = cos acos b+ sin asin b to what you wrote in part (a). Then, simplify using Even- 3. Using cos ( a b), let s find sin ( a b) +. π a) From Fundamental Identities Day 1, recall we can use the Co-Function Identity: sinθ = cos θ to sin a+ b. Let θ = 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 ( ) π sinθ = cos θ again. cos a b = cos acos b+ sin asin b to what you wrote in part (b). 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 ( a+ b) = 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. π π π π a) sin cos cos sin 5 7 5 7 b) tan19 + tan 41 1 tan19 tan 41 π π tan ( ) tan ( ) 3 c) π π 1+ tan ( ) tan ( ) 3 d) cos 6 cos 94 sin 6 sin 94. Use a sum or difference identity to find the exact value for each function. a) cos( 75 ) b) sin ( 195 ) π c) cos 1 d) 11π tan 1
3. Simplify the following expressions as much as possible: a) sinx b) tan 6 4 4. Prove the following identities sin x y sin xy sin x cos y a) cos x y cos xy cos x cos y b) c) tan x tan x tan x
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. Use the function shown to answer the following questions. a) Write a tangent function that fits the graph. b) Notice this graph could also be the graph of the parent function. c) Use identities to PROVE your answer from part a is the same as the parent function in part b.
Questions 7-8 review concepts covered in Unit 5 and extend these concepts to Unit 6 7. Use the given information to evaluate the identity. HINT: you do NOT need to find the angles first!! a) If 3 sin A = when A is in Quadrant I and if 5 cos 5 B = when B is in Quadrant IV, evaluate cos( A B) 13. b) If 5 tan x = when x is in Quadrant III and if 1 cos y 3 = when y is in Quadrant II, evaluate sin ( x y) 5 +. 8. Write each trigonometric expression as an algebraic expression. a) sinarcsin x arccos x b) 1 cos sin x tan 1 x In preparation for lesson 5.4 9. Prove the following identities HINT: Start with the left side. 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. π π. 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 if necessary. tan A 9. sina 10. sin x cotxsin x 1 tan A
sinx 11. cot x 1. sinxcotxtan x 1 cosx 13. cscxsecx cscx 14. cos4x1 8sin xcos x 3 15. sin3u 3cos usin u sin u 16. cos 3x cos x cos x cos x
PreCalculus PreRequisites for Solving Trig Equations Graph the equation y = sin x and y 1 on your calculator. 1. How many times do these two graphs intersect?. On the interval 1 0,, how many solutions does the equation sin x have? Find them in terms of. 3. Find the solutions to the following equations on the interval 0, a sketch of the parent function may help you determine how many solutions each equation has. a) cos x b) sin x 1 c) tan x 1 1 sin x e) cos x f) cos x 0 d) 3 g) tan x 0 h) sin x i) tan x 3 Factor the following trigonometric expressions. 4. cos x cos x 1 5. 1sin x sin x 6. sin x sin xcos x 7. 4 cos x 8. sec x sec x 9. cos x5cos x 7
Pre Calculus Worksheet: Solving Trigonometric Equations For questions 1-6, solve each equation on the interval [0, π). 1. cos x + 5 = 4. sin xtan x sin x= 0 3. cos x+ sin x= 1 4. sin x 5sin x+ = 0 5. cos x cos x 0 + = 6. cos x= cos x+ 1 7. Solve for x on the domain [0, π ) : 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. 4cos x 3 = cos x 11. sin x sin x 3 = 1. sin x 3sin x 0 + = 13. cos x = 1 14. cos x 4 7 cos = x 15. sin x sin xcos x= 0 16. 3sin x cos x 0 = 17. 3π sin x = 1