Determine missing sides using trigonometry

The learner determines an unknown side length in a right triangle from one known side and an acute angle by interpreting the opposite, adjacent, and hypotenuse sides relative to that angle and selecting sine, cosine, or tangent. They rearrange the resulting trigonometric equation, use degree-mode calculator values, and round appropriately while checking that the result is consistent with the triangle’s geometry; this scope excludes non-right triangles and the laws of sines and cosines.

Detailed Explanation: Determine missing sides using trigonometry

In a right triangle, choose the given acute angle first. Then name the sides relative to that angle:

Opposite: across from the angle
Adjacent: next to the angle, but not the hypotenuse
Hypotenuse: across from the right angle; it is always the longest side

Use SOHCAHTOA to choose a trigonometric ratio:

\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}},\qquad \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}},\qquad \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}

Example

Find the missing side length $x$ .

Suppose a right triangle has:

an acute angle of $35^\circ$ ,
an adjacent side of length $8$ ,
an opposite side of length $x$ .

Since the known side and unknown side are the adjacent and opposite sides, use tangent:

\tan(35^\circ)=\frac{x}{8}

Multiply both sides by $8$ to isolate $x$ :

x=8\tan(35^\circ)

Make sure your calculator is in degree mode, then calculate:

x\approx 8(0.7002)

x\approx 5.6

Therefore, the missing side is approximately

\boxed{5.6\text{ units}}

The answer makes sense because the $35^\circ$ angle is less than $45^\circ$ , so the opposite side should be shorter than the adjacent side. Indeed, $5.6<8$ .

Learn by doing: Determine missing sides using trigonometry

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Trigonometry - Solve Side Lengths from Values

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