Cos ( α ) = 11/8 and sin ( β ) = √ 7/5 . Find cos ( α + β )

Step-by-step explanation:

We are given that:

Where both α and β are in QI.

And we want to find cos(α + β).

First, let's determine the side lengths for each angle.

For α, we are given that its cosine is √(11)/8.

And since cosine is the ratio of the adjacent side to the hypotenuse, the adjacent side to α is √11 and the hypotenuse is 8.

Therefore, the opposite side will be:

Hence, for α, the adjacent side is √11, the opposite side is √53, and the hypotenuse is 8.

Likewise, for β, we are given that its sine is √7/5.

And since sine is the ratio of the opposite side to the hypotenuse, the adjacent side of β is:

In summary:

For α, the adjacent is √11, the opposite is √53, and the hypotenuse is 8.

For β, the adjacent is 3√2, the opposite is √7, and the hypotenuse is 5.

Using an angle addition identity, we can rewrite our expression as:

And since both α and β are in QI, all trig ratios will be positive.

Using the above information, we can substitute in the following values:

Finally, simplify: