Question Video: Solving a Trigonometric Equation Using Double-Angle Identities Mathematics

Find all the solutions of the equation sin 𝑥/2 cos 𝑥/2 = 1/2 for 0 ≤ 𝑥 ≤ 90°. Write the answer approximated to the nearest whole angle in degrees.

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Video Transcript

Find all the solutions of the equation sin 𝑥 over two cos 𝑥 over two equals one-half for 𝑥 greater than or equal to zero and less than or equal to 90 degrees. Write the answer approximated to the nearest whole angle in degrees.

In order to answer this question, we recall one of the double-angle identities. This states that sin of two 𝜃 is equal to two sin 𝜃 cos 𝜃. By letting 𝜃 be the angle in this question 𝑥 over two, we have sin 𝑥 is equal to two sin 𝑥 over two cos 𝑥 over two. Dividing through by two, we have sin 𝑥 over two cos 𝑥 over two is equal to one-half sin 𝑥. Substituting this into the given equation, we have a half sin 𝑥 is equal to one-half. And multiplying through by two, sin 𝑥 is equal to one.

Next, we recall the graph of 𝑦 equals sin 𝑥 for values of 𝑥 between zero and 360 degrees as shown. In this question, we are told that 𝑥 lies between zero and 90 degrees inclusive. And since the sin of 90 degrees is equal to one, we can conclude that 𝑥 is equal to 90 degrees.

The only solution to the equation sin 𝑥 over two cos 𝑥 over two equals one-half, where 𝑥 is greater than or equal to zero and less than or equal to 90 degrees, is 𝑥 is equal to 90 degrees. And since the solution is an integer, we do not need to approximate our answer.

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