Video: Using Sum and Difference of Angles Identities to Solve Trigonometric Equations Involving Special Angles

Find the solution set for 𝑥 given sin 𝑥 cos 35 + cos 𝑥 sin 35 = (√2)/2 where 0° < 𝑥 < 360°

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Video Transcript

Find the solution set for 𝑥 given sin 𝑥 cos 35 plus cos 𝑥 sin 35 is equal to root two over two where 𝑥 is between zero and 360 degrees.

One of the compound angle trigonometrical identities states that sin of 𝐴 plus 𝐵 is equal to sin 𝐴 cos 𝐵 plus cos 𝐴 sin 𝐵. In our question, 𝐴 is equal to 𝑥 and 𝐵 is equal to 35. We can therefore rewrite the equation sin 𝑥 cos 35 plus cos 𝑥 sin 35 is equal to root two over two as sin of 𝑥 plus 35 is equal to root two over two.

Taking the inverse sine of both sides of this equation gives us 𝑥 plus 35 is equal to inverse sin of root two over two. The inverse sin of root two over two is equal to 45 degrees. Therefore, 𝑥 plus 35 is equal to 45. We could solve this equation by subtracting 35 from both sides to find one solution of the equation. However, we were asked for the solution set which suggests there will be more than one solution.

We could find all the other solutions between zero and 360 degrees either by drawing a sine graph or using the CAST diagram as shown. There will be one solution in the A quadrant and one solution in the S quadrant. These will be symmetrical about the 𝑦-axis. The solution in the first quadrant between zero and 90 is 45 degrees as we have already found. The solution in the second quadrant is 135 degrees, as 180 minus 45 equals 135.

We can find more solutions by adding 360 degrees to each of these answers. This is because the sine graph continues indefinitely. However, these net solutions of 405 degrees and 495 degrees will lie outside of the range required as we are asked to find solutions between zero and 360 degrees. This means that 𝑥 plus 35 can either be equal to 45 degrees or 135 degrees.

Subtracting 35 from both sides of this equation will give us our solution set for 𝑥. 45 minus 35 is equal to 10. And 135 minus 35 is equal to 100. This means that the solution set for the equation sin 𝑥 cos 35 plus cos 𝑥 sin 35 equals root two over two. Our 𝑥 equals 10 degrees. And 𝑥 equals 100 degrees.

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