Verify Trigonometric Identities

Transcription

1 4.3 a., A..A; P..C TEKS Verify Trigonometric Identities Before You graphed trigonometric functions. Now You will verify trigonometric identities. Why? So you can model the path of Halley s comet, as in Ex. 4. Key Vocabulary trigonometric identity Recall from Lesson 3.3 that if an angle u is in standard position with its terminal side intersecting the unit circle at (x, y), then x cos u and y sin u. Because (x, y) is on a circle centered at the origin with radius, it follows that: x y cos u sin u The equation cos u sin u is true for any value of u. A trigonometric equation that is true for all values of u (in its domain) is called a trigonometric identity. Several fundamental trigonometric identities are listed below, some of which you have already learned. y r (cos u, sin u) (x, y) u x KEY CONCEPT For Your Notebook Fundamental Trigonometric Identities Reciprocal Identities csc u sin u sec u cos u cot u tan u Tangent and Cotangent Identities tan u sin u cos u cot u cos u sin u Pythagorean Identities sin u cos u tan u sec u cot u csc u Cofunction Identities sin p u cos u cos p u sin u tan p u cot u Negative Angle Identities sin (u) sin u cos (u) cos u tan (u) tan u You can use trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and verify other identities. 94 Chapter 4 Trigonometric Graphs, Identities, and Equations

2 E XAMPLE Find trigonometric values Given that sin u 4 and p < u < p, find the values of the other five trigonometric functions of u. Solution STEP Find cos u. sin u cos u Write Pythagorean identity. 4 cos u Substitute 4 for sin u. cos u 4 Subtract 4 from each side. cos u 9 Simplify. REVIEW TRIGONOMETRY For help with finding the sign of a trigonometric function value, see p STEP cos u 6 3 cos u 3 Take square roots of each side. Because u is in Quadrant II, cos u is negative. Find the values of the other four trigonometric functions of u using the known values of sin u and cos u. tan u sin u cos u cot u cos u sin u csc u sin u 4 4 sec u cos u 3 3 E XAMPLE Simplify a trigonometric expression Simplify the expression tan p u sin u. tan p u sin u cot u sin u Cofunction identity cos u sin u (sin u) Cotangent identity cos u Simplify. E XAMPLE 3 Simplify a trigonometric expression Simplify the expression csc u cot u. sin u csc u cot u csc u cot u csc u sin u Reciprocal identity csc u (csc u ) csc u Pythagorean identity csc 3 u csc u csc u Distributive property csc 3 u Simplify. 4.3 Verify Trigonometric Identities 9

3 GUIDED PRACTICE for Examples,, and 3 Find the values of the other five trigonometric functions of u.. cos u 6, 0 < u < p. sin u 3 7, π < u < 3p Simplify the expression. 3. sin x cot x sec x 4. tan x csc x sec x. cos p u sin (u) VERIFYING IDENTITIES You can use the fundamental identities on page 94 to verify new trigonometric identities. When verifying an identity, begin with the expression on one side. Use algebra and trigonometric properties to manipulate the expression until it is identical to the other side. E XAMPLE 4 Verify a trigonometric identity Verify the identity sec u sec u sin u. sec u sec u sec u sec u sec u Write as separate fractions. sec u Simplify. cos u Reciprocal identity sin u Pythagorean identity E XAMPLE Verify a trigonometric identity Verify the identity sec x tan x cos x. sin x sec x tan x cos x tan x Reciprocal identity cos x sin x cos x Tangent identity sin x cos x Add fractions. VERIFY IDENTITIES To verify the identity, you must introduce sin x into the denominator. Multiply the numerator and the denominator by sin x so you get an equivalent expression. sin x cos x p sin x sin x Multiply by sin x sin x. sin x cos x ( sin x) Simplify numerator. cos x cos x ( sin x) Pythagorean identity cos x sin x Simplify. 96 Chapter 4 Trigonometric Graphs, Identities, and Equations

4 E XAMPLE 6 Verify a real-life trigonometric identity SHADOW LENGTH A vertical gnomon (the part of a sundial that projects a shadow) has height h. The length s of the shadow cast by the gnomon when the angle of the sun above the horizon is u can be modeled by the equation below. Show that the equation is equivalent to s h cot u. s h sin (908u) sin u Solution Simplify the equation. s h sin (908u) sin u Write original equation. h sin p u sin u Convert 908 to radians. h cos u sin u Cofunction identity h cot u Cotangent identity GUIDED PRACTICE for Examples 4,, and 6 Verify the identity. 6. cot (u) cot u 7. csc x ( sin x) cot x 8. cos x csc x tan x 9. (tan x )(cos x ) tan x 4.3 EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS on p. WS for Exs.,, and 4 TAKS PRACTICE AND REASONING Exs. 9, 4, 4, 43, 44, 46, and 47 MULTIPLE REPRESENTATIONS Ex. 4. VOCABULARY What is a trigonometric identity?. WRITING What does the cofunction identity sin p u about the graphs of y sin x and y cos x? cos u tell you EXAMPLE on p. 9 for Exs. 3 9 FINDING VALUES Find the values of the other five trigonometric functions of u. 3. sin u 3, 0 < u < p 4. tan u 3 7, 0 < u < p. cos u 6, 3p < u < π 6. sin u 7 0, π < u < 3p 7. cot u, p < u < π 8. sec u 9 4, p < u < π 4.3 Verify Trigonometric Identities 97

5 9. TAKS REASONING If csc u 3 and p < u < π, what is the value of tan u? A Ï B Ï 3 3 C Ï 3 3 D Ï EXAMPLES and 3 on p. 9 for Exs. 0 4 SIMPLIFYING EXPRESSIONS Simplify the expression. 0. sin x cot x. sin (u) cos (u). csc u sin u cot u 3. cos u ( tan u) 4. tan p x. cos p x csc x cos p u csc u cos u 7. sin p u sec u 8. cos x cot x sec x sin x cos p x sec x 0. csc x cot x sin (x) cot x. cos x tan (x) cos x ERROR ANALYSIS Describe and correct the error in simplifying the expression.. 3. sin u ( cos u) cos u cos u tan (x) csc x sin x cos x p sin x cos x sec x 4. TAKS REASONING Which of the following is the simplified form of the expression cos u sec u? A tan u B C D sin u EXAMPLES 4 and on p. 96 for Exs. 34 VERIFYING IDENTITIES Verify the identity.. sin x csc x 6. tan u csc u cos u 7. cos p u 8. sin sin (u) p x tan x sin x 9. csc u cot u sin u sec u 30. cos u sin u 3. sin x cos x cot x csc x 3. sin (x) tan x cos x 33. cos x sin x sin x cos x csc x 34. sin x cos (x) csc x cot x 3. ODD AND EVEN FUNCTIONS A function f is odd if f(x) f(x). A function f is even if f(x) f(x). Which of the six trigonometric functions are odd? Which are even? VERIFYING IDENTITIES Verify the identity. 36. ln sec u ln cos u 37. ln tan u ln sin u ln cos u 38. CHALLENGE Use the Pythagorean identity sin u cos u to derive the other Pythagorean identities, tan u sec u and cot u csc u. 98 WORKED-OUT SOLUTIONS on p. WS TAKS PRACTICE AND REASONING MULTIPLE REPRESENTATIONS

6 PROBLEM SOLVING EXAMPLE 6 on p. 97 for Exs RATE OF CHANGE In calculus, it can be shown that the rate of change of the function f(x) sec x cos x is given by this expression: sec x tan x sin x Show that the expression for the rate of change can be written as sin x tan x. 40. PHYSICAL SCIENCE Static friction is the amount of force necessary to keep a stationary object on a flat surface from moving. Suppose a book weighing W pounds is lying on a ramp inclined at an angle u. The coefficient of static friction u for the book can be found using this equation: uw cos u W sin u a. Solve the equation for u and simplify the result. b. Use the equation from part (a) to determine what happens to the value of u as the angle u increases from 08 to 908. u 4. MULTIPLE REPRESENTATIONS The path of Halley s comet is an ellipse with the sun as a focus. The path can be estimated by the equation below, where r is the comet s distance (in astronomical units) from the sun and u is the angle (in radians) between the horizontal major axis and the comet. Sun r u Halley s comet r sin p u a. Writing an Equation Simplify the equation given above. b. Drawing a Graph Use a graphing calculator to graph the equation from part (a). c. Making a Table Make a table of values for the equation from part (a) in which u starts at 0 and increases in increments of p. Use the table to 4 approximate the closest and farthest distance, in miles, Halley s comet is from the sun. (Note: astronomical unit ø 93 million miles.) 4. TAKS REASONING Use a reciprocal identity to describe what happens to the value of sec u as the value of cos u increases. On what intervals does this happen? 43. TAKS REASONING Use the tangent identity to describe what happens to the value of tan u as the value of sin u increases and the value of cos u decreases. On what intervals does this happen? 4.3 Verify Trigonometric Identities 99

7 44. TAKS REASONING When light traveling in a medium (such as air) strikes the surface of a second medium (such as water) at an angle u, the light begins to travel at a different angle u. This change of direction is defined by Snell s law, n sin u n sin u, where n and n are the indices of refraction for the two mediums. Snell s law can be derived from the equation: u n u n n Ï cot u n Ï cot u a. Derive Simplify the equation to derive Snell s law: n sin u n sin u. b. Solve If u 8, u 38, and n, what is the value of n? c. Interpret If u u, what must be true about the values of n and n? Explain when this situation would occur. 4. CHALLENGE Brewster s angle is the angle u, at which light reflected off water is completely polarized, so that glare is minimized when you look at the water with polarized sunglasses. Brewster s angle can be found using Snell s law (see Exercise 44). a. Let sin u n n sin u and cos u n n cos u. Add the two equations to show that n sin u n cos u. n n b. Show that the equation from part (a) can be simplified to n n n sin u n n cos u. n c. Solve the equation from part (b) to find Brewster s angle: u tan n n MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 4.; TAKS Workbook 46. TAKS PRACTICE Which equation will produce the widest parabola when graphed? TAKS Obj. A y 3x B y x C y.x D y x REVIEW Skills Review Handbook p. 988; TAKS Workbook 47. TAKS PRACTICE Reflect nrst in the line x. In which quadrant will the image of point R appear? TAKS Obj. 7 F Quadrant I H Quadrant III G Quadrant II J Quadrant IV R S T 3 4 x 3 4 y 930 EXTRA PRACTICE for Lesson 4.3, p. 03 ONLINE QUIZ at classzone.com