Would like to know

Question

Find an angle θθ with 0∘<θ<360∘0∘<θ<360∘ that has the same:

Sine as 190∘: θ=

Cosine as 190∘: θ=

Answer 1

A 190 degree angle on the unit circle would put it in the 3rd quadrant with a 10 degree reference angle. Since sine is negative in the 3rd quadrant, we need to find another angle that has a 10 degree reference angle whose sine ratio is also negative. The other quadrant where sine is negative is the 4th quadrant, and a 10 degree reference angle in the 4th quadrant would equate to a 350 degree angle, so sin 190 = sin 350.

Following the same type of reasoning for cos 190, this would also involve a 10 degree reference angle in the 3rd quadrant where cosine, like sine, is negative. But between 0 and 360, the other quadrant where cosine is negative is the 2nd quadrant, so a 2nd-quadrant angle with a 10 degree reference angle would be 170 degrees, so cos 170 = cos 190.