Concept 3

Solve right-triangle problems using trigonometric ratios; use special right triangles.

Use special right-triangle ratios first, then fall back on sine, cosine, or tangent.

Core Idea

Use special right-triangle ratios first when you see $45 - 45 - 90$ or $30 - 60 - 90$ . If the triangle is not special, identify opposite, adjacent, and hypotenuse relative to the angle and use sine, cosine, or tangent.

Understanding

Rule: A $45 - 45 - 90$ triangle has side ratio $1 : 1 : \sqrt 2$ , and a $30 - 60 - 90$ triangle has ratio $1 : \sqrt 3 : 2$ . Recognizing those patterns is faster than rebuilding the triangle from scratch.

If the triangle is not special, use SOHCAHTOA carefully. The side names depend on the marked angle, so opposite and adjacent can change when the reference angle changes. The hypotenuse never changes because it is always opposite the right angle.

Question

Worked Example

In a right triangle, one angle measures $30 \circ$ and the hypotenuse is $10$ . What is the length of the side opposite the $30 \circ$ angle?

Select an answer to see the explanation