For the given right triangle, we want to write the ratios for the sine, cosine, and tangent of $∠ M .$ Let's start by identifying the $hypotenuse$ of the triangle and the sides that are $opposite$ and $adjacent$ to $∠ M .$

We see that the length of the hypotenuse is $4 .$ The length of the side adjacent to $∠ M$ is $2$ and the length of the side opposite to $∠ M$ is $23 .$ With this information, we can find the desired ratios.

Ratio	Definition	Value
$sin M$	$\frac{Length of leg opposite ∠ M}{Length of hypotenuse}$	$\frac{2 3}{4} = \frac{3}{2}$
$cos M$	$\frac{Length of leg adjacent ∠ M}{Length of hypotenuse}$	$\frac{2}{4} = \frac{1}{2}$
$tan M$	$\frac{Length of leg opposite ∠ M}{Length of leg adjacent ∠ M}$	$\frac{2 3}{2} = 3$

Extra

Reciprocal Ratios

Later in the book we will learn that the sine, cosine, and tangent ratios each have a reciprocal ratio. The reciprocal ratios are cosecant, secant, and cotangent. These terms are often shortened to csc, sec, and cot, respectively.

csc θ = \frac{hypotenuse}{opposite}, sec θ = \frac{hypotenuse}{adjacent} cot θ = \frac{adjacent}{opposite}

In the given right triangle, the length of the hypotenuse is

4 .

The length of the side adjacent to

∠ M

is

2

and the length of the side opposite to

∠ M

is

23 .

With this information, we can also find the reciprocal ratios of the ratios that we found in the exercise.

Ratio	Definition	Value
$csc M$	$\frac{Length of hypotenuse}{Length of leg opposite ∠ M}$	$\frac{4}{2 3} = \frac{2 3}{3}$
$sec M$	$\frac{Length of hypotenuse}{Length of leg adjacent ∠ M}$	$\frac{4}{2} = 2$
$cot M$	$\frac{Length of leg adjacent ∠ M}{Length of leg opposite ∠ M}$	$\frac{2}{2 3} = \frac{3}{3}$

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