Using trigonometric ratios and a calculator, the unknown side lengths and angle measures 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 is 3 units long, and the side adjacent to is 12 units long. So to find angle , we write an equation using tangent: . To find the measure of angle , we ask the calculator, “What angle has a tangent of ?” To ask that, we use arctangent by writing . The arctangent of a positive number is the measure of an acute angle whose tangent is that number. If we know the cosine of an angle, we use arccosine to look up the angle measure (the arccosine of a number between 0 and 1 is the measure of an acute angle whose cosine is that number). And if we know the sine of an angle, we use arcsine (the arcsine of a number between 0 and 1 is the measure of an acute angle whose sine is that number). So , which means angle measures about 14 degrees.

Angle can be calculated using another trigonometric equation or the Triangle Angle Sum Theorem. Let's use arctangent again. We know , so , which is about 76 degrees. This matches the answer we get with the Triangle Angle Sum Theorem: .