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Chapter 7 Section 7.6 Trigonometric Equations If a more complicated angle is inside the sine or cosine there is one more step of solving for x at the end. The example to the right is how to solve: i.e. c 2 2 cos(5 x ) Solutions: 5x 3 2k 4 and 5x 5 2k 4 2 2 5x 135 1. Draw unit circle. cos(5x) 2 2 2. Draw horizontal or vertical line the correct distance on x or y axis. 3 4 2 2 3. Find angles where line hits the unit circle. 2 2 4. Add 2k to each angle. 5 4 5. Solve for x. k 360 and 5 x 225 k 360 3 2k 5 2k and x x 27 k 72 and x 45 k 72 20 5 20 5 The Equations: a sinx + b = c and a cosx + b = c x If the sine or cosine is not isolated (i.e. all by itself on one side of the equation) carry out the algebra to isolate the sine or cosine. Now apply what we did above to get the solutions. 2 cos(5 x) 2 0 Solve: 2 cos(5 x) 2 0 2 cos(5 x) 2 cos(5 x) 2 2 Solutions: x 3 2k 20 5 and x 5 2k 20 5 x 27 k 72 and x 45 k 72 Give answers in radians. Regroup: Simplifying Equations Before Solving Sometimes equations might need to be simplified using a combination of trigonometric identities and algebra before solving them. Factor: Factor: 2 sin 𝜃 − 1 2 cos 𝜃 + 1 = 0 Solve each equation 2𝜋 3 −1 2 2 cos 𝜃 + 1 = 0 2 cos 𝜃 = −1 −1 cos 𝜃 = 2 𝜃 = 2𝜋 + 2𝑘𝜋 3 𝜃 = 4𝜋 3 + 2𝑘𝜋 4𝜋 3 Solve: 2 sin x sin x 2 2 sin x sin x 0 sin x2 sin x 1 0 Equations with Powers of Sine or Cosine 2 sin x 0 and If the equation you are trying to solve has a power of sine or cosine, set one side equal to zero and factor the other side. Use what was just discussed to solve the parts you get. Solve: 2 sin x 1 0 sin x 0 and sin x 5 6 cos 2 2x 4 cos 2x 3 0 cos 2x 3cos 2x 1 0 cos 2x 3 0 and cos 2x 1 0 cos 2x 3 and cos 2x 1 1 2 6 0 No Solutions (first equation) Solutions: Solutions: 2k x 0 2k and x 2k x 2 x 6 2k and x 56 2k x 2 4k Rearrange Square both sides Regroup Apply Identity Cancel Solve: 5 sin 2 x 14 cos x 13 0 Equations With Both Sine and Cosine 5(1 cos 2 x) 14 cos x 13 0 If a trigonometric equation has both a sine and cosine in it use trigonometric identities to change it to an equation involving either all sine or all cosine. 5 cos x 14 cos x 8 0 36.8699 5 5 cos 2 x 14 cos x 13 0 323.13 2 5 cos x 14 cos x 8 0 (cos x 2)(5 cos x 4) 0 2 cos x 2 0 and cos x 2 and No Solution and Find the other angle. 360-36.8699=323.13 5 cos x 4 0 cos x 4 5 x cos 1 4 5 The solutions for this are: x 36.8699 x 36.8699 k 360 x 323.13 k 360