### Home > PC > Chapter 7 > Lesson 7.3.2 > Problem7-113

7-113.

Given $\sec A = \frac{5}{3}$ and $\cot B = \frac { 12 } { 5 }$ where $0 \lt A$, $B < \frac { \pi } { 2 }$, find the exact value of each trig expression.

1. $\sin A$

Draw a reference triangle in Quadrant 1 for both angles.

Label the 3rd side.
Both are pythagorean triples: $3,4,5$ and $5,12,13$

Find sin A from the triangle sides.

$\frac{4}{5}$

1. $\cos B$

Find $\cos B$ from the triangle sides.

$\frac{12}{13}$

1. $\sin\left(A + B\right)$

Use the sum and difference formula:
sin(A + B) = sinAcosB + cosAsinB

$\text{sin}(A+B)=\frac{4}{5}\cdot \frac{12}{13}+\frac{3}{5}\cdot \frac{5}{13}=$

$\frac{63}{65}$

1. $\cos\left(A + B\right)$

Use the sum and difference formula:
$\cos\left(A + B\right) = \cos A \cos B − \sin A \sin B$

1. $\tan\left(A + B\right)$

$\text{tan}(A+B)=\frac{\text{sin}(A+B)}{\text{cos}(A+B)}$