Video: Using Subtraction Formula for Cosine Function to Solve a Trigonometric Equation Involving Special Angles

Find the solution set of sin π‘₯ cos 16 βˆ’ cos π‘₯ sin 16 = √(2)/2, where 0Β° < π‘₯ < 360Β°.

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Video Transcript

Find the solution set of sin π‘₯ multiplied by cos 16 minus cos π‘₯ multiplied by sine 16 equals root two divided by two, where π‘₯ lies between zero and 360 degrees.

In order to solve this problem, we’ll use the identity that sin of 𝐴 minus 𝐡 is equal to sin 𝐴 cos 𝐡 minus cos 𝐴 sin 𝐡. In our example, 𝐴 is equal to π‘₯ and 𝐡 is equal to 16. We can therefore rewrite the equation as sine of π‘₯ minus 16 equals root two divided by two.

The angle in the first quadrant whose sine is equal to root two divided by two is 45 degrees, and the angle in the second quadrant whose sine is equal to root two divided by two is 135 degrees. This is because sine of 45 is equal to root two divided by two, and sine of 135 is equal to root two divided by two.

This means that sine of π‘₯ minus 16 is equal to sine 45, and sine of π‘₯ minus 16 is equal to sine of 135. Solving the first equation gives us π‘₯ minus 16 is equal to 45. Adding 16 to both sides of this equation gives us π‘₯ equals 61 degrees.

Solving the second equation gives us π‘₯ minus 16 equals 135. Adding 16 to both sides of this equation gives us π‘₯ equals 151 degrees. This means that the solution set of sin π‘₯ multiplied by cos 16 minus cos π‘₯ multiplied by sin 16 equals root two divided by two is 61 degrees and 151 degrees.

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