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Solving Trigonometric Equations Involving Multiple Angles 6.3 JMerrill, 2009 Strategies for Solving Trig. Equations with Multiple Angles • If the equation involves functions of 2x and x, transform the functions of 2x into functions of x by using identities • If the equation involves functions of 2x only, it is usually better to solve for 2x directly and then solve for x • Be careful not to lose roots by dividing off a common factor • Remember: You can always graph to check your solutions Example • Solve cos 2x = 1 – sin x for 0 ≤ x < 2π cos2x 1 sin x 1 2 sin2 x 1 sin x 1 2 sin2 x 1 sin x 0 sin x (2 sin x 1) 0 2 sin2 x sin x 0 sin x 0 x 0, 1 sin x 2 5 x , 6 6 You Do • Solve for 0o≤θ<360o cos 2x = cos x 2cos2 x 1 cos x 0 2cos2 x cos x 1 0 (2cos x 1)(cos x 1) 0 1 cos x ,cos x 1 2 120o , 240o , 0o Example • Solve 3cos2x + cos x = 2 for 0 ≤ x < 2π 3cos2x cos x 2 3(2cos2 x 1) cos x 2 6cos2 x 3 cos x 2 6cos x cos x 5 0 2 (6cos x 5)(cos x 1) 0 5 cos x cos x 1 6 x 0.59, 5.70 x 3.14 All of the previous examples were solved for x. Now we’ll solve for 2x directly. Example • Solve 2sin2x = 1 for 0o ≤ θ < 360o 2 sin 2x 1 Pretend the 2 isn’t in front of the x and solve it (solve sin x = ½ ) 1 sin 2x 2 2x 30 ,150 ,390 ,510 o o o x 15o , 75o ,195o ,255o 0 You Do • Solve for 0o≤θ<360o tan22x-1=0 tan 2x 1 2 tan2x 1 2x 45o ,135o ,225o ,315o , 405o , 495o ,585o ,675o , x 22.5o ,67.5o ,112.5o ,157.5o , 202.5o ,247.5o ,292.5o ,337.5o ,