Video: Using the Addition Formula for a Cosine Function to Evaluate a Trigonometric Expression

Find the solution set for 𝑥 given cos 𝑥 cos 2𝑥 − sin 𝑥 sin 2𝑥 = 1/2 where 0° < 𝑥 < 360°.

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Video Transcript

Find the solution set for 𝑥 given cos 𝑥 cos two 𝑥 minus sin 𝑥 sin two 𝑥 is equal to a half where 𝑥 is between zero and 360 degrees.

One of our compound angle trigonometrical identities states that cos of 𝐴 plus 𝐵 is equal to cos 𝐴 cos 𝐵 minus sin 𝐴 sin 𝐵. In this question, our 𝐴 is equal to 𝑥 and our 𝐵 is equal to two 𝑥. The equation cos 𝑥 cos two 𝑥 minus sin 𝑥 sin two 𝑥 equals a half can therefore be rewritten as cos of 𝑥 plus two 𝑥 is equal to a half.

𝑥 plus two 𝑥 is equal to three 𝑥. Therefore, cos of three 𝑥 equals one-half. Taking the inverse cos, or cos to the minus one, of both sides of this equation gives us three 𝑥 is equal to cos to the minus one of one-half. Typing the right-hand side into our calculator gives us 60. Therefore, three 𝑥 is equal to 60 degrees.

Solving this equation by dividing both sides by three will give us one solution. However, we were asked to find the solution set which suggests there is more than one answer. We can find the other solutions either by drawing the cosine graph or using CAST.

This tells us that three 𝑥 will be quarter to two values, one in the A quadrant and one in the C quadrant. The diagram will be symmetrical about the 𝑥-axis. This means that as our first solution for three 𝑥 was equal to 60 degrees, our second solution will be equal to 300, as 360 minus 60 is equal to 300. We can therefore say the two solutions for the equation are that three 𝑥 is equal to 60 or three 𝑥 is equal to 300.

As the cosine graph continues indefinitely, these solutions will repeat every 360 degrees. For example, adding 360 to 60 gives us 420 degrees. So, three 𝑥 could also be equal to 420. Likewise, 300 plus 360 is equal to 660. So, a fourth possible value is that three 𝑥 equals 660 degrees. Continuing this process gives us further values of 780 degrees and 1020 degrees. This means that three 𝑥 could also be 780 or 1020. In fact, this process will carry on forever. However, we were asked for solutions of 𝑥 between zero and 360 degrees.

As these are all the values for three 𝑥, we need to divide each of them by three to calculate the corresponding value for 𝑥. 60 divided by three is equal to 20. And 300 divided by three is equal to 100. Dividing the other four values by three gives us answers of 140, 220, 260, and 340 degrees.

There are six angles in the solution set between zero and 360 degrees for the equation cos 𝑥 cos two 𝑥 minus sin 𝑥 sin two 𝑥 equals one-half. They are 𝑥 equal 20 degrees, 100 degrees, 140 degrees, 220 degrees, 260 degrees, and 340 degrees. We could check each of these answers individually by substituting the angles into the equation. Each of them will give us an answer of one-half.

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