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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.
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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.
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In this question, our 𝐴 is equal to 𝑥 and our 𝐵 is equal to two 𝑥.
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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.
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𝑥 plus two 𝑥 is equal to three 𝑥.
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Therefore, cos of three 𝑥 equals one-half.
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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.
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Typing the right-hand side into our calculator gives us 60.
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Therefore, three 𝑥 is equal to 60 degrees.
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Solving this equation by dividing both sides by three will give us one solution.
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However, we were asked to find the solution set which suggests there is more than one answer.
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We can find the other solutions either by drawing the cosine graph or using CAST.
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This tells us that three 𝑥 will be quarter to two values, one in the A quadrant and one in the C quadrant.
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The diagram will be symmetrical about the 𝑥-axis.
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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.
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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.
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As the cosine graph continues indefinitely, these solutions will repeat every 360 degrees.
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For example, adding 360 to 60 gives us 420 degrees.
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So, three 𝑥 could also be equal to 420.
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Likewise, 300 plus 360 is equal to 660.
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So, a fourth possible value is that three 𝑥 equals 660 degrees.
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Continuing this process gives us further values of 780 degrees and 1020 degrees.
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This means that three 𝑥 could also be 780 or 1020.
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In fact, this process will carry on forever.
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However, we were asked for solutions of 𝑥 between zero and 360 degrees.
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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.
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60 divided by three is equal to 20.
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And 300 divided by three is equal to 100.
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Dividing the other four values by three gives us answers of 140, 220, 260, and 340 degrees.
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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.
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They are 𝑥 equal 20 degrees, 100 degrees, 140 degrees, 220 degrees, 260 degrees, and 340 degrees.
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We could check each of these answers individually by substituting the angles into the equation.
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Each of them will give us an answer of one-half.