Select an appropriate trigonometric ratio

The learner identifies the given acute angle and matches the available side relationships to sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), or tangent (opposite/adjacent), recognizing that “opposite” and “adjacent” depend on the chosen reference angle. This supports setting up right-triangle proportions to determine unknown sides or angles; inverse trigonometric functions, non-right triangles, and generalized trigonometric identities are beyond this scope.

Detailed Explanation: Select an appropriate trigonometric ratio

To choose a trigonometric ratio, first focus on the given acute angle. Then name the sides from the angle’s point of view:

Opposite: the side directly across from the angle
Adjacent: the side next to the angle that is not the hypotenuse
Hypotenuse: the longest side, always opposite the right angle

Use the ratio that includes the sides you know and the side you want:

\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}

\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}

\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}

Example

A right triangle has an angle of $40^\circ$ . The side next to the $40^\circ$ angle has length $x$ , and the hypotenuse has length $15$ . Find $x$ .

Step 1: Identify the sides relative to the $40^\circ$ angle.

Adjacent side: $x$
Hypotenuse: $15$

Step 2: Choose the ratio.

The available sides are adjacent and hypotenuse, so use cosine:

\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}

Step 3: Substitute the known values.

\cos(40^\circ)=\frac{x}{15}

Step 4: Solve for $x$ .

Multiply both sides by $15$ :

x=15\cos(40^\circ)

Using a calculator:

x\approx 11.5

So, the missing side is approximately

\boxed{11.5}

Always identify the sides in relation to the chosen angle before selecting sine, cosine, or tangent.

Learn by doing: Select an appropriate trigonometric ratio

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Trigonometry - Solve Side Lengths from Diagrams

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