Find all solutions A between 0 and 360 without a calculator: (a) cosA = cos251 (b) cosA = 1.5 (c) sin A = sin 220 (d) cos A = cos(−110)

Answer 1

First of all, I'm going to assume that all angles are in measures of degrees. Secondly, I'm going to assume you made a typo in question b) and that it should say cos(A) = cos(1.5). If it is really cos(A) = 1.5 then there are no angles that have a s=cosine larger than 1.

To understand these problems you must understand the mapping of sine and cosine in the unit circle (a circle with radius 1 in the coordinate plane):

Because the triangle has has a hypotenuse of 1, we can write relationships for sine and cosine of angle A like this;

cos(A) = adjacent/hypotenuse = x/1 therefore x = cos(A)

sin(A) = opposite/hypotenuse = y/1 therefore y = sin(A)

Because of this, we can also name (x, y), the coordinate of a point on the circle, (cos(A), sin(A)). The cos(A) is the x-coordinate of the point and sin(A) is the y-coordinate of the point.

Knowing that, if we find the angle 251°, which is 71° past 180°. we can see that it has the same x-coordinate as an angle that is 71° less than 180°:so to get the answer to a), just do 180 - 71 = 109° so A = both 251° and 109° (This case is shown in blue below)

For question b), the cosine of 1.5˜ would e the same as 358.5° (both have the same x-coordinate) so for question b) A = 1.5 and A = 358.5 (shown in red below)

Answer 2

a) cosine 251 is negative (in QIII). Reference angle is 71 degrees. cosine is also negative in QII. Reference angle of 71 in QII would equal 180-71 = 109 degrees. so A = 109 and 251.

b) A is the null set as cosine only ranges between -1 and 1

c) sin 220 is negative (in QIII). Reference angle is 40 degrees. sine is also negative in QII. Reference angle of 40 in QIV would equal 360-40 - 320 degrees. so A = 220 and 340.

c) cos(-110) = cos(250). cosine 250 is negative (in QIII). Reference angle is 70 degrees. cosine is also negative in QII. Reference angle of 70 in QII would equal 180-70 = 110 degrees. so A = 110 and 250.