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.