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1 TRIG-Fall 2011-Jordan Trigonometry, 9th edition, Lial/Hornsby/Schneider, Pearson, 2009 Chapter 5: Trigonometric Identities Section 5.1 Fundamental Identities Negative-Angle Identities sin (-θ) = - sin θ csc (-θ) = -csc θ cos (-θ) = cos θ sec (-θ) = sec θ tan (-θ) = -tan θ cot (-θ) = -cot θ Reciprocal Identities sin 1 csc csc 1 sin cos 1 sec sec 1 cos tan 1 cot cot 1 tan Pythagorean Identities Quotient Identities sin2 cos2 1 tan sin cos tan2 1 sec2 cot cos sin 1 cot 2 csc2 Example 1 values. a) sec θ b) sin θ c) cot (-θ) If tan θ = -5/3 and θ is in quadrant II, find each of the following function 2 Example 2 Express sec x in terms of sin x Example 3 First rewrite cot 2 θ(1 + tan 2 θ) in terms of sin θ and cos θ and then simplify so that no quotients appear in the final expression. 3 Section 5.2 Verifying Trigonometric Identities Verifying Trigonometric Identities An identity is an equation that is true for all its domain values. To verify an identity, we show that one side of the identity can be rewritten in a form that is identical to the other side. Guidelines for Verifying Trigonometric Identities Know the fundamental identities and their equivalent forms, e.g., sin2 x + cos2 x = 1 is equivalent to cos2 x = 1 – sin2 x. Start working with the more complicated side of the equation and try to turn it into the simpler side. Perform any indicated operations such as adding fractions, squaring binomials, distributing, or factoring. Sometimes it is helpful to express all trigonometric functions on one side of an equation in terms of sine and cosine. Sometimes it is helpful to rewrite one side of the equation in terms of a single trigonometric function. Multiplying both the numerator and denominator of a fraction by the same factor (usually the conjugate of the numerator or denominator) may help. As you selection substitutions, keep in mind the side you are not changing. It represents your goal. Look for the identity or function which best links the two sides. If you get really stuck, abandon the side you’re working on and start working on the other side. Try to make the two sides “meet in the middle.” Caution: Verifying identities is not the same as solving equations. Techniques used in solving equations, such as adding the same term to both sides or multiplying both sides by the same factor, are not valid when verifying identities. Example 1 Perform the indicated operation and simplify the result: cos sin sin 1 cos Example 2 Factor the expression: cot 4 x 3 cot 2 x 2 csc sec cot Example 3 Simplify: Example 4 Verify the identity: 1 csc x sin x sec x tan x Example 5 Verify the identity: 1 1 2 csc2 x 1 cos x 1 cos x Example 6 Verify the identity: sin x csc x cot x 1 cos x 4 5 Section 5.3 Sum and Difference Identities for Cosine Cosine of a Sum or Difference cos( A B) cos A cos B sin A sin B cos( A B) cos A cos B sin A sin B Caution: cos (A + B) cos A + cos B, etc. Example 1 5 a) cos 12 Find the exact value. b) cos 95 cos 35 + sin 95 sin 35 Example 2 Given sin s = 24/25 and cos t = -4/5, s in Quadrant II and t in Quadrant III, find a) cos (s + t) b) cos (s – t) Cofunction Identities sin (90 - ) = cos θ csc (90 - ) = sec θ cos (90 - ) = sin θ sec (90 - ) = csc θ tan (90 - ) = cot θ cot (90 - ) = tan θ If θ is in radian measure, then replace 90° with π/2. Example 3 Prove the cofunction identity cos (90 - ) = sin θ Example 4 Write sin 5 in terms of the cofunction of a complementary angle. 8 6 Section 5.4 Sum and Difference Identities for Sine and Tangent Sine of a Sum or Difference sin( A B) sin A cos B cos Asin B sin( A B) sin A cos B cos Asin B Caution: sin (A + B) sin A + sin B, etc. Tangent of a Sum or Difference tan( A B) Example 1 a) sin b) tan A tan B 1 tan A tan B tan( A B) tan A tan B 1 tan A tan B Find the exact value. 12 tan(105) c) sin 90° cos 135° - cos 90° sin 135° Example 2 Use an identity to write sin as a single function of θ. 6 Example 3 sin s = 4/5 and cos t = -5/13, s in quadrant II and t in quadrant III a) Find sin (s + t) b) Find tan (s + t) c) Find the quadrant of s + t 7 Section 5.5 Double-Angle Identities Double-Angle Identities Caution: sin 2A ≠ 2 sin A Example 1 Given a) find sin θ or cos 2 A cos A , etc. A and θ terminates in quadrant III, b) find cos θ Example 2 Given a) find sin 2θ and sin θ < 0, b) find cos 2θ c) find tan 2θ Example 3 Use an identity to write value or as a single number. Example 4 Verify the following identity: as a single trigonometric function 8 Product-to-Sum Identities Caution: sin M sin N ≠ sin (MN), etc. Example 5 functions. Write the following expression as a sum or difference of trigonometric 2 cos 85° sin 140° Sum-to-Product Identities Caution: sin M – sin N ≠ sin (M – N), etc. (You cannot factor out or distribute a trig function) Example 6 Write the following expression as the product of two functions. cos 2θ – cos 4θ 9 Section 5.6 Half-Angle Identities Half-Angle Identities Caution: The plus or minus signs of sin (A/2) and cos (A/2) are selected according to the quadrant in which A/2 terminates. Example 1 Find the exact value of cos 112.5° Example 2 Use an identity to write the following expression as a single trigonometric function. Example 3 Given that sin θ = -7/25 with 180° < θ < 270°, a) find the exact value of sin (θ/2) b) find the exact value of cos (θ/2) c) find the exact value of tan (θ/2)