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Trigonometric Equations and Simplification

Learn how to simplify trigonometric expressions and solve trigonometric equations step by step.

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Trigonometric Equations and Simplification

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  1. cos (x + π) = cosx cosπ– sinx sinπ EXAMPLE 3 Simplify an expression Simplify the expression cos (x + π). Sum formula for cosine = (cos x)(–1) – (sinx)(0) Evaluate. = – cosx Simplify.

  2. Solve sin ( x + ) + sin ( x – ) = 1 for 0 ≤ x < 2π. ANSWER π π 3 3 In the interval 0 ≤ x <2π, the only solution isx = . sin ( x + ) + sin ( x – ) = 1 1 1 π π sinx cos+ cosx sin+ sinx cos– cosx sin 2 2 π π π π π 3 3 = 1 2 3 3 3 3 sinx + cos x + sinx – cosx = 1 3 3 2 2 = 1 sinx EXAMPLE 4 Solve a trigonometric equation Write equation. Use formulas. Evaluate. Simplify.

  3. π t π t 182 182 = –6cos ( ) + 12.1 = 2 sin ( – 1.35) + 12.1 EXAMPLE 5 Solve a multi-step problem Daylight Hours The number h of hours of daylight for Dallas, Texas, and Anchorage, Alaska, can be approximated by the equations below, where tis the time in days and t = 0 represents January 1. On which days of the year will the two cities have the same amount of daylight? Dallas: h1 Anchorage: h2

  4. 2 sin ( – 1.35) + 12.1 = – 6 cos ( ) + 12.1 sin ( – 1.35) = – 3 cos ( ) = – 3 cos ( ) sin ( ) cos 1.35– cos ( ) sin1.35 = – 3 cos ( ) sin ( ) (0.219)– cos ( ) (0.976) 0.219 sin ( ) = – 2.024 cos ( ) π t π t π t π t π t π t π t π t π t π t π t π t 182 182 182 182 182 182 182 182 182 182 182 182 EXAMPLE 5 Solve a multi-step problem SOLUTION Solve the equation h1 = h2 for t. STEP 1

  5. tan ( ) – 1.463+ nπ – 84.76 + 182n 97, or on April 8 – 84.76 + 182(1) π t π t π t – 84.76 + 182(2) 279, or on October7 182 182 182 EXAMPLE 5 Solve a multi-step problem = – 9.242 = tan–1 (– 9.242)+ nπ t STEP 2 Find the days within one year (365 days) for which Dallas and Anchorage will have the same amount of daylight. t t

  6. ANSWER ANSWER ANSWER sinx cosx tanx for Examples 3, 4, and 5 GUIDED PRACTICE Simplify the expression. 6. sin (x + 2π) 8. tan (x – π) 7.cos (x – 2π)

  7. ANSWER about5.65 π t π t 75 75 for Examples 3, 4, and 5 GUIDED PRACTICE 9. Solve 6 cos ( ) + 5 = – 24 sin ( + 22) + 5 for 0 ≤ t < 2π.

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