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DATE 5-2 PERIOD NAME DATE 5-2 Study Guide and Intervention Verifying Trigonometric Identities You can use a graphing calculator to test whether an equation might be an identity by graphing the functions related to each side of the equation. If the graphs of the related functions do not coincide for all values of x for which both functions are defined, the equation is not an identity. If the graphs appear to coincide, you can verify that the equation is an identity by using trigonometric properties and algebraic techniques. sec2 x − 1 = sin2 x. Verify that − 2 sec x The left-hand side of this identity is more complicated, so start with that expression first. 2 sec x (tan2 x + 1) − 1 sec x tan2 x =− 2 sec x ( Simplify. ) Quotient Identity and Reciprocal Identity sin θ 1 - sin2 θ =− sin x =− · cos2 x 2 2 sin θ cos2 θ =− sin θ cos θ =− · cos θ sin θ Simplify. cos x = sin x # Multiply. Notice that the verification ends with the expression on the other side of the identity. 2 1. sec θ - cos θ = sin θ tan θ ( cos θ ) 1 − cos2 θ sin2 θ sin θ 1 sec θ − cos θ = − − cos θ = − =− = sin θ − cos θ cos θ cos θ = sin θ tan θ 2. sec θ = sin θ(tan θ + cot θ) ( cos θ ) cos θ sin θ sin θ (tan θ + cot θ) = sin θ − +− = sin θ ( cos θ sin θ ) 1 = sin θ − 1 =− = sec θ sin θ ( sin2 θ + cos2 θ − cos θ sin θ ) cos θ 3. tan θ csc θ cos θ = 1 ( cos θ )( sin θ ) sin θ 1 − tan θ csc θ cos θ = − cos θ = 1 θ − cot θ − = sec2 θ 4. csc 2 2 2 Chapter 5 005-032_PCCRMC05_893806.indd 10 10 [-2π, 2π] scl: π by [-4, 4] scl: 1 2 Pythagorean Identity Factor cos2 θ. Rewrite in terms of cot θ using a Quotient Identity. [-2π, 2π] scl: π by [-4, 4] scl: 1 2 Exercises Test whether each equation is an identity by graphing. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. 1. sin x + cos x cot x = csc x 2. 2 - cos2 x = sin2 x π , Y1 = 1.5 When x = − sin x + cos x cot x 4 ( sin x ) cos x = sin x + cos x − and Y2 = 0.5; therefore, the equation is not an identity. sin2 x + cos2 x sin x cos x =− = sin x + − 2 sin x 1 =− = csc x 1 − sin θ (cot2 θ + 1) − cot2 θ csc2 θ − cot2 θ 1 − = −− =− = sec2 θ cos2 θ cos2 θ 1 − sin2 θ sin x Glencoe Precalculus Chapter 5 3/13/09 12:10:25 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Verify each identity. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A4 = cot θ cos θ # Exercises Rewrite using a common denominator. 005-032_PCCRMC05_893806.indd 11 11 Glencoe Precalculus 3/13/09 12:10:29 PM Answers (Lesson 5-2) = Example Use a graphing calculator to test whether csc θ - sin θ = cot θ cos θ is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. The equation appears to be an identity because the graphs of the related functions coincide. Verify this algebraically. 1 csc θ - sin θ = − - sin θ Rewrite in terms of sine using a Reciprocal Identity. Pythagorean Identity sin2 x − cos2 x − 1 − cos2 x (continued) Identifying Identities and Nonidentities To verify an identity means to prove that both sides of the equation are equal for all values of the variable for which both sides are defined. sec x − 1 − = − 2 2 Study Guide and Intervention Verifying Trigonometric Identities Verify Trigonometric Identities Example PERIOD Lesson 5-2 Chapter 5 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE 5-2 PERIOD NAME DATE 5-2 Practice Word Problem Practice Verifying Trigonometric Identities Verifying Trigonometric Identities Verify each identity. csc x 1. − = cos x sin y − 1 1 − sin x sin y + 1 sin y + 1 − sin y + 1 csc x sin x cos x = − · − − cos x cot x + tan x sin x sin x cos x 1 1 − − = − − sin y − 1 sin y + 1 sin2 y − 1 − +− cos x sin x = 1. SURVEYING From a point h meters above level ground, a surveyor measures the angle of depression θ of the corner of a building lot. The distance d from the corner to the point on the ground directly under the surveyor’s instrument can be described by the equation d = h cot θ. 1 1 2. − −− = −2 sec2 y cot x + tan x cos x − cos2 x + sin2 x = cos x = cos x # =− 1 2 − -cos2 y ( 1 - tan θ ) cot θ - 1 Verify that d = h − is an cos θ 3. sin3 x - cos3 x = (1 + sin x cos x)(sin x - cos x) 4. tan θ + − = sec θ 1 + sin θ )( ) 1 + cos θ sin2 θ + 1 + 2 cos θ + cos2 θ sin θ − + − = −− 1 + cos θ sin θ sin θ (1 + cos θ) (1 − sin θ)2 1 − sin θ = − 2 + 2 cos θ sin θ (1 + cos θ) 2 (1 + cos θ) sin θ (1 + cos θ) = − 2 (1 − sin θ)2 1 − sin θ = −− = − 1 + sin θ (1 + sin θ)(1 − sin θ) 2 =− = 2 csc θ # # sin θ Test whether each equation is an identity by graphing. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. 1 + sin x cos x cos x 7. − =− 1 − sin x 8. sin x(sec x + cot x) = cos x π When x = − , Y = 1.7 and Y2 = 0.7; 4 1 1 + sin x 1 − sin x 1 + sin x 1 − sin x cos x (1 + sin x) = − 1 − sin2 x cos x (1 + sin x) = − cos2 x 1 + sin x = − # cos x cos x cos x − =− ·− therefore, the equation is not an identity. Glencoe Precalculus 9. PHYSICS The work done in moving an object is given by the formula W = Fd cos θ, where d is the displacement, F is the force exerted, and θ is the angle between the displacement and the force. Verify that Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. = − csc θ Fd cot θ − csc θ Fd = Chapter 5 Answers 005-032_PCCRMC05_893806.indd 12 cos θ (− sin θ ) − 1 − sin θ = Fd cos θ − sin θ · sin θ = Fd cos θ 12 5θ − 6(sec θ − tan θ sin θ) ) ( cos θ cos θ sin θ 1 −− = 5θ − 6(− cos θ ) cos θ 1 − sin θ = 5θ − 6( − cos θ ) cos θ = 5θ − 6 (−) cos θ sin θ 1 = 5θ − 6 − −− · sin θ 2 ) 2 = 5θ − 6 cos θ # 2. CAROUSEL The angular velocity ω of each person riding a carousel is the number of degrees, or radians, that a point on the edge of the carousel moves in a unit of time. You can use the equation y = r sin (60ω), where r is the radius of the circle, to sketch a graph representing the position of a person on the carousel with respect to a stationary point every 60 seconds. - 4. TELESCOPE The largest working refracting telescope is housed at the University of Chicago’s Yerkes Observatory. In a refracting telescope, as the diameter d of the lens increases, its resolution θ can be improved, which means the telescope can better separate images in the sky. The relationship between resolution θ in degrees and diameter d in meters can be expressed by ω 1.22λ sin θ = − , where λ is the wavelength d S of light in meters, or by csc θ + 1 1.22λ − =− . Verify that 1 + csc θ (− sin θ ) a. Verify that the equation y = r tan (60ω) cos (60ω) is an equivalent formula. 2 1 + csc θ (− sin θ ) 2 r tan (60ω) cos (60ω) = r sin (60ω) − cos (60ω) csc θ + 1 3/13/09 12:10:35 PM 005-032_PCCRMC05_893806.indd 13 csc θ + 1 csc θ + csc θ − = − 2 · cos (60ω) ( ) 1 + csc θ − sin2 θ = r sin (60ω) Chapter 5 d csc θ + 1 sin θ = − is an identity. csc θ + 1 = − csc θ (csc θ +1) 1 = sin θ # =− csc θ graphs should appear to coincide. Glencoe Precalculus ) 2 b. How could you use a graphing calculator to check your answer in part a? Graph each function. The Fd cot θ W=− is an equivalent formula. W= ) cos θ ( cos θ = h cot θ # = h( − sin θ ) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A5 ( cos θ - sin θ − cos θ sin θ 1-− θ - sin θ cos θ − = h cos · − sin θ cos θ - sin θ (sec θ - tan θ)2 = sec2 θ - 2 sec θ tan θ + tan2 θ 1 − 2 sin θ + sin2 θ cos θ sin θ =h − 1 + cos θ sin θ 6. − + − = 2 csc θ 1 + cos θ sin θ = −− 2 cos θ - sin θ − cos θ − -1 sin θ − 13 Glencoe Precalculus 3/13/09 12:10:41 PM Answers (Lesson 5-2) ( cos θ sin θ cos θ tan θ + − = − + − cos θ 1 + sin θ 1 + sin θ sin θ + sin2 θ + cos2 θ = − = (sin x - cos x)(1 + sin x cos x) (cos θ)(1 + sin θ) 1 + sin θ = (1 + sin x cos x)(sin x - cos x) # = − = sec θ # (cos θ)(1 + sin θ) sin2 θ sin θ +− − cos θ cos2 θ ( ( 1 - tan θ ) = h cot θ - 1 d=h − sin3 x - cos3 x = (sin x - cos x) (sin2 x + sin x cos x + cos2 x) 1 1 =− −2 − cos θ cos2 θ 3. TRAINS The equation y = 5θ - 6 cos θ can be used to help describe the movement of a train’s wheel as the train moves forward. Use a graphing calculator to test whether the equation 5θ - 6 cos θ = 5θ - 6 (sec θ -tan θ sin θ) is an identity. If it appears to be an identity, verify it. If not, find a value for which both sides are defined but not equal. equivalent equation. = -2 sec2 y # 1 − sin θ 5. (sec θ - tan θ)2 = − 1 + sin θ PERIOD Lesson 5-2 Chapter 5 NAME