Remark
If the two angles are complimentary they add up to 90°. So the first job is to find the third angle. In the diagram below, the third angle is labeled C, so you must find it first.
After that, you need to use the Pythagorean Theorem to solve for QC which at this point is not known.
After completing that, You need to find the cosine of P and Q so that they can be added together.
Step One
Find angle C
All triangles have 180°
P + Q are complementary. Given
P + Q = 90° Definition
P + Q + C = 180 Property of a triangle.
90 + C = 180 Subtract 90 from both sides.
C = 180 - 90
C = 90°
Step Two
Find QC
When you have a right triangle the Pythagorean Theorem can be used to find any missing side.
Sin(Q) = opposite / hypotenuse [See diagram]
Sin(Q) = 4/5 Given
a = 4
b = ??
c = 5
4^2 + b^2 = 5^2
16 + b^2 = 25 Subtract 16 from both sides.
b^2 = 25 - 16
b^2 = 9
Take the square root of both sides.
sqrt(b^2) = sqrt(9)
b = 3
Step 3
Define Cos(Q) and Cos(P)
Cos(Q) = adjacent side (3) over the hypotenuse (5)
Cos(Q) = 3/5
Cos(P) has the same basic definition. The adjacent side is 4 for angle P
Cos(P) = 4/5
Step Four
Add the two findings from Step 3
Cos(Q) + Cos(P) = 3/5 + 4/5 = 7/5 <<<<< Answer
