Question

For positive acute angles A and B, it is known that cos A= 60/61 and tan B= 45/28. Find the value of sin(A+B) in simplest form

Answer 1

`\displaystyle \sin(A+B)=(3008)/(3233)`

Explanation:

We are given that:

`\displaystyle \cos(A)=(60)/(61)\text{ and } \tan(B)=(45)/(28)`

And that the two angles A and B are both in QI (positive acute angles).

Given this, we want to determine the value of sin(A + B).

Recall that cosine is the ratio of the adjacent side to the hypotenuse. Using this information, we can determine the opposite side with respect to A:

`\displaystyle o=√(61^2-60^2)=√(121)=11`

Likewise, for B, tangent gives the ratio of the opposite side to the adjacent. Using this information, we can determine the hypotenuse of B:

`h=√(45^2+28^2)=√(2809)=53`

In conclusion:

With respect to A, the adjacent side is 60, the opposite side is 11, and the hypotenuse is 61.

With respect to B, the adjacent side is 28, the opposite side is 45, and the hypotenuse is 53.

We can rewrite our original expression as:

`\sin(A+B)=\sin(A)\cos(B)+\sin(B)\cos(A)`

And since A and B are both in QI, all trigonometric ratios will be positive.

Using the above conclusions, substitute:

`\displaystyle \sin(A+B)=\left((11)/(61)\right)\left((28)/(53)\right)+\left((45)/(53)\right)\left((60)/(61)\right)`

Simplify. Hence:

`\displaystyle \sin(A+B)=(308+2700)/(3233) = (3008)/(3233)`