# Inverse Sine, Cosine and Tangent

The inverse of SOHCAHTOA

The inverse trigonometric functions (sin-1, cos-1, and tan-1) allow you to find the measure of an angle in a right triangle. All that you need to know are any two sides as well as how to use SOHCAHTOA.

### Inverse SOHCAHTOA Way vs Interior Angle Sum

Compare This Method To the tried and true theorem that the sum of the interior angles of a triangle is 180°.

What is the degree measure of LNM?

Since the total measure of the interior angles of a triangle is 180 degrees we can verify the measure of LNM:
180° -16° - 90° =74 °

Alternately, you could use the inverse of one of the SOHCAHTOA functions, in this case the inverse of sine (sin-1)! To find, an angle of a right triangle all that we need to know is the length of two sides! Then use the same SOHCAHTOA ratios -- just in a different fashion See the example below.

### YouTube Vid: How to Calculate Inverse SOHCAHTOA

A good video on how to use your a TI-Graphing Calculator to calculate the inverse sine,cosine or tangent.

### Example Problem

To find the measure of $\angle CAB$, the shaded angle, consider the sides that we know.

#### Compare sine with inverse sine.

$sin(b)$
$$sin (b) = \frac{ac}{ab}$$ Sine of angle outputs the ratio of 2 sides of a triangle
$sin^{-1}(b)$
$$sin^{-1} \left( \frac{ac}{ ab} \right) = m\angle ABC$$ Inverse sine of 2 sides outputs an angle measurement

### Practice Problems

##### Problem 1

Use inverse sine, cosine or tangent to calculate the measure of the shaded angle on the left.

tan-1(24/18) = 53°
tan-1(5/12) =23°
##### Problem 3
sin-1(36/39) = 67°
##### Problem 4

Since you know all 3 sides, you could use any of the following:

= sin-1(7/25) = 16.3°
= cos-1(6/15) = 16.3°
= tan-1(7/24) = 16.3°
##### Problem 5

Since you know all 3 sides, you could use any of the following:

= sin-1(8/10) = 53.13°
= cos-1(6/10) = 53.13°
= tan-1(8/6) = 53.13°