Pre Calculus Worksheet 5.5 Solve each triangle using the Law of Sines. 1. ABC where B16, C 103, c 1. WXY where X 81, Y 59, w 9 3. How can you tell, without using the Law of Sines, that a triangle cannot be formed by using the measurements A89, B 104, a 8? Explain. 4. Use PQR at the right to answer the questions below. a) Draw the altitude to PR and label it QT. 10cm Q b) Write an expression for the length of QT. Do not find the actual length. P 17cm R c) Find the area of PQR. 5. To find the length of the span of a proposed ski lift from A to B, a surveyor measures the angle DAB to be 5 and then walks off a distance of 1000 feet to C and measures the angle ACB to be 15. What is the distance from A to B? B D 5 A 1000 ft C 6. An emergency dispatcher must determine the position of a caller reporting a fire. Based on the caller s cell phone records, she is located in the area shown. Overcome by the desire to solve for any missing lengths, the dispatcher momentarily forgets about the fire and wants to know what the unknown side lengths are in the triangle. Tower 60.1 Tower 1 67.5 5.4 4.6 mi Tower 3
7. US 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay? 140 Pelican Bay mi 1/8 mi 1/8 mi 135 US 41 8. A buoy is anchored offshore to mark a sandbar. The straight shoreline at that location runs north and south. From two observation points on the shore.4 miles apart, the bearings to the buoy are 134 and. a) What is the distance from the buoy to each of the observation points? b) How far is the buoy from the shore? 9. An adventurer who is stuck on the top of a cliff is trying to decide whether or not he can jump to the next ledge. In his free time he is able to determine the angle from the bottom of the tree to the edge of the cliff and the angle from the top of the tree to the edge of the cliff as shown in the diagram below. While he was climbing the tree to measure that second angle he figured out that the tree is 13 feet tall. Find the distance d the adventurer will have to jump in order to make it to the ledge (assuming he doesn t trip over the roots of the tree and fall to the bottom of the cliff). 38 d 15
Pre Calculus Worksheet 5.6 Solve each triangle using the Law of Cosines. 1. ABC where B131, a13, c 8. WXY where x 8, y 17, w 30 Determine whether to use SohCahToa, Law of Sines or Law of Cosines to solve triangle DEF. Explain. 3. f 15, D31, E 4 4. d 3, e4, f 13 5. D8, E 98, d 6 6. f 3, e5.5, D40 7. d 11, D, E 68 8. f 7, e0, E 119 9. Give two reasons why you cannot have a triangle with sides 13 cm, 9 cm and 4 cm. 10. Find the area of the triangle in question 6. 11. Find the area of the triangle in question 4. 1. In attempting to fly from city A to city B, an aircraft followed a course that was 10 in error, as indicated in the figure. After flying a distance of 50 miles, the pilot corrected the course by turning at point C and flies the remaining distance. If the straightline distance from A to B is 119 miles, and the average speed of the aircraft was 50 miles per hour, how much time was lost due to the error? B A 10 C
13. US 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay? 140 Pelican Bay mi 1/8 mi 1/8 mi 135 US 41 14. An airplane flies north from Ft. Myers to Sarasota a distance of 150 miles, and then changes his bearing to 50 and flies to Orlando, a distance of 100 miles. a) How far is it from Ft. Myers to Orlando? b) What bearing is needed for the pilot to return from Orlando to Ft. Myers? 15. Solve the following equation for P: p = h + d - hdcos( P) 17. A streetlight is designed as shown below. Determine the angle in the design. 3 θ 4.5
Pre Calculus Worksheet 5.5 (ambiguous case) 1. Explain why given SSA is called the ambiguous case. State whether the given measurements determine zero, one or two triangles ABC.. C 10, a 18, c 9 3. C 36, a 17, c 16 4. B 8, b 17, c 15 5. For any questions 4 that have TWO triangles, solve BOTH triangles. Solve the triangle WXY with the given parts below. IF there are two triangles, SOLVE BOTH!! 6. Y 103, w 46, y 61 7. X 57, w 11, x 10 8. On Spring Break, Bob and his friends decide to go 4-wheeling off road in his new Jeep. The Jeep has a winch (a lifting device with a cable) that is used to pull the Jeep in case it gets stuck. After driving too fast over a hill, Bob finds himself stuck in the middle of a shallow stream. While wading through the stream to attach the cable to a tree a hill on the other side of the stream (see diagram), Bob ponders the mathematics of his situation He wonders what the angle of elevation of the cable was before his Jeep was pulled to the edge of the stream. 100 ft 37 65 ft
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 1. tan x sec x csc x. csc cot sec Strategy II: Pythagorean or Co-Function Identities 3. sin x cos x csc x cot x 4. sin tan cos cos Strategy III: Factoring the GCF sin x 5. sin x sin xcos x 6. 3 cot w cot cos w w Strategy IV: Even-Odd Identities 7. cot( x) cot x 8. 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 9. 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!! Use extra paper to show your work!!! Verify (Prove) each identity. 10. tan x = sin x + sin x tan x 11. sin x cos x sec x cos x 1. tan xcsc x 1 sec x 13. (sin x)(tan x cos x cot x cos x) = 1 cos²x 14. cos = cot sin csc 15. 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 4. 4 4 sin cos Strategy VII: Factoring trinomials with box 4 4 5. 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. i 5 5 5 5
Let s try it with trigonometric expressions. Strategy VIII: Multiply by 1 6. cos x 1 sinx HINT: Multiply by 1 sinx in numerator and denominator. 7. tan x sec x 1 Verify (Prove) each identity. Use extra paper as necessary!! 8. 1 1 cotxcscx sec x1 sec x1 9. sec x sin x sin x cos x cot x 10. cos 11. sin csc csc sin 13cos 4cos 14 cos u u u u sin u 1 cos 1. cos csc sin sinsec 3 cot 13. sin sin csc 1cos 1cos 14. 1 tan tan 15. 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 Day 1 Use the sum or difference identities to prove each identity. Use extra paper as necessary!! 1. 3. 3 cos sin 3 tan u cot u. cos xsin x 4. sin x sin 3xcos x cos3xsin x Write the expression as the sine, cosine or tangent of a single angle. Then, evaluate if possible. 5. sin cos cos sin 5 7 5 7 6. tan19tan 41 1 tan19tan41 tan tan 3 7. 1 tan tan 3 8. cos 6cos94sin 6sin 94 Use the sum and difference identities to find the exact value for each function. 9. cos75 10. sin 195 11. cos 1 1. 11 tan 1
Now let s get a little deeper into where these identities come from We first have to start with the assumption that cos ab cos acos b sin asin b. This identity CAN be derived just ask! 13. Using cosa b, let s find cosa b. 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 Even-Odd Identities. cos ab cos acos b sin asin b to what you wrote in part (a). Then, simplify using 14. 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 rewrite the sine of an angle as a cosine function. Apply this identity to sin a b. Let a b. 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 15. 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 Day 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?. Use a sum or difference identity to find an exact value (NO CALCULATOR). p a) sin( - ) b) sin( 55 ) 1 7p c) cos( ) d) tan (-105) 1 3. Simplify the following expressions as much as possible: p p a) sin( x + ) = b) cos( x ) 6 - = 4 p c) tan( ) q + = 4 5. Prove the following identities use a separate sheet of paper. 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
6. Prove the following identities: Hint: 4x = 3x + x AND x = 3x x use a separate sheet of paper tan 4u- tan u a) sin 4x + sin x= sin 3xcos x b) tan 5utan 3u= 1 - tan 4utan u 7. 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 use a separate sheet of paper. 8. Write each trigonometric expression as an algebraic expression. a) sin( arcsin x arccos x) - + b) ( 1 - cos cos x - sin 1 x) - c) ( 1 - sin tan x cos 1 x) - d) cos( arcsin x - arctan x) 9. Prove the following identities use separate sheet of paper. a) sin( x) = sin xcos x b) ( ) cos cos sin x = x- x tanx = 1 - tan x c) cos( x) = 1- sin x d) tan( 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
PreCalculus PreRequisites for Solving Trig Equations Factor the following polynomials using any and all factoring techniques from Algebra 1 and Algebra. 1. 3. 4x 4x 1. 3x y 6xy 4. x 3x 9x 4 Factor the following trigonometric expressions using the same techniques from above. 5. cos x cos x 1 6. 1 sin x sin x 7. sin x sin xcos x 8. 4 cos x 9. sec x sec x 10. cos x 5cos x 7 Solve each of the following equations for x. 11. x 4x 0 1. (x + )(x 1) = 0 13. x x 3 14. 3x x Simplify each of the following expressions: 15. 1 cos 16. sin -1 ( 1) 17. arctan 3 18. arctan( 1) 19. arccos 0 0. 1 3 cos 1. 1 1 sin. cos 1 1 3. Graph the equation y = sin x and y 1 on your calculator. a) How many times do these two graphs intersect? 1 b) On the interval 0,, how many solutions does the equation sin x have? Find them in terms of. c) Find the solutions to the following equations on the interval 0, use your calculator to help you determine how many solutions each equation has. i) cos x ii) sin x 1 iii) tan x 1 iv) 1 sin x v) cos x vi) cos x 0 3 vii) tan x 0 viii) sin x ix) tan x 3
Pre Calculus Worksheet: Solving Trigonometric Equations 1. Solve: x sin 1 1. Solve for x on the domain [0, ) : 1 sin x 3. Solve the system with your graphing calculator: y sin x 1 y 4. Explain the difference in your solutions for questions 1-3. For questions 5 10, solve each equation on the interval [0, ). 5. cosx 5 4 6. sin xtan x sin x 0 7. cos x sin x 1 8. sin x 5sin x 0 9. cos x cosx 0 10. cos x cosx 1
For questions 11 18, find all solutions to each equation. 11. 4cosx 3 cosx 1. sin x sin x 3 13. sin x 3sin x 0 14. cos x 1 15. cos x 4 7cos x 16. sin x sin xcos x 0 17. 3sin x cos x 0 18. 3 sin x 1