Lesson 9

A triangle with side lengths 3, 4, and 5 is a right triangle by the converse of the Pythagorean Theorem. What are the measures of the acute angles?

Find all missing side and angle measures.

Using trigonometric ratios and a calculator, the missing sides and angles of right triangles can be found.

Using the right triangle table we can estimate angle measures as in previous lessons. However, with a calculator, we can find angles more precisely.

The side opposite angle $A$ is 3 units long, and the side adjacent to $A$ is 12 units long. So to find angle $A$, we write an equation using tangent: $\tan(\alpha) = \frac{3}{12}$. To find the measure of angle $A$ we ask the calculator, “What angle has a tangent of $\frac{3}{12}$?” To ask that, we use arctangent by writing $\arctan \left(\frac{3}{12} \right)$. If we know the cosine, we use arccosine to look up the angle, and if we know the sine, we use arcsine. So $\alpha=\arctan \left( \frac{3}{12} \right)$, which means angle $A$ measures about 14 degrees.

Angle $B$ can be calculated using another trigonometric equation or the Triangle Angle Sum Theorem. Let's use arctangent again. We know $\tan(\theta)=\frac{12}{3}$, so $\theta=\arctan \left( \frac{12}{3} \right)$, which is about 76 degrees. This matches the answer we get with the Triangle Angle Sum Theorem: $180-90-14 = 76$.