Question

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

Answer 1

`\displaystyle \cos(\alpha+\beta)=(3√(22)-√(371))/(40)`

Explanation:

We are given that:

`\displaystyle \cos(\alpha)=(√(11))/(8)\text{ and } \sin(\beta)=(\sqrt7)/(5)`

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:

`o=\sqrt{8^2-(√(11))^2}=√(53)`

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:

`a=\sqrt{5^2-(√(7))^2}=√(18)=3√(2)`

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:

`\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)`

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:

`\displaystyle \cos(\alpha +\beta)=\Big((√(11))/(8)\Big)\Big((3\sqrt2)/(5)\Big)-\Big((√(53))/(8)\Big)\Big((\sqrt7)/(5)\Big)`

Finally, simplify:

`\displaystyle \cos(\alpha +\beta)=(3√(22))/(40)-(√(371))/(40)=(3√(22)-√(371))/(40)\approx -0.1298`