Exercise 12 Page 514

For the given right triangle, we want to write the ratios for the sine, cosine, and tangent of $∠ A$ and $∠ B .$

Ratios for $∠ A$

Let's start by identifying the $hypotenuse$ of the triangle and the sides that are $opposite$ and $adjacent$ to $∠ A .$

We see that the length of the hypotenuse is $78 .$ The length of the side adjacent to $∠ A$ is $72$ and the length of the side opposite to $∠ A$ is $30 .$ With this information, we can find the desired ratios.

Ratio	Definition	Value
$sin A$	$\frac{Length of leg opposite ∠ A}{Length of hypotenuse}$	$\frac{3 0}{7 8} = \frac{5}{1 3}$
$cos A$	$\frac{Length of leg adjacent ∠ A}{Length of hypotenuse}$	$\frac{7 2}{7 8} = \frac{1 2}{1 3}$
$tan A$	$\frac{Length of leg opposite ∠ A}{Length of leg adjacent ∠ A}$	$\frac{3 0}{7 2} = \frac{5}{1 2}$

Ratios for $∠ B$

We already know the length of the $hypotenuse$ is $78 .$ Let's identify the sides that are $opposite$ and $adjacent$ to $∠ B .$

The length of the side adjacent to $∠ B$ is $30$ and the length of the side opposite to $∠ B$ is $72 .$ With this information, we can find the desired ratios.

Ratio	Definition	Value
$sin B$	$\frac{Length of leg opposite ∠ B}{Length of hypotenuse}$	$\frac{7 2}{7 8} = \frac{1 2}{1 3}$
$cos B$	$\frac{Length of leg adjacent ∠ B}{Length of hypotenuse}$	$\frac{3 0}{7 8} = \frac{5}{1 3}$
$tan B$	$\frac{Length of leg opposite ∠ B}{Length of leg adjacent ∠ B}$	$\frac{7 2}{3 0} = \frac{1 2}{5}$

Exercise 12 Page 514

Ratios for ∠A

Ratios for ∠B

Ratios for $∠ A$

Ratios for $∠ B$