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.

sin-1(73.24/76.19) = 74°

YouTube Vid: How to Calculate Inverse SOHCAHTOA

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

Want to buy a TI Graphing Calculator ?

Compare sine  with inverse sine.

inverse sine, SOHCAHTOA General Difference: sine is the ratio of two actual sides of a right triangle (the opposite & hypotenuse) sin(B) = AC/AB

Inverse or sin-1 is an operation that uses the same two sides of a right triangle as sine does (opposite over hypotenuse) in order to find the measure of the angle (in this case b) sin-1(AC/AB) = measure of angle B

Key difference: Although both sine and inverse sine involve  the opposite side and hypotenuse of a right triangle, the result of these two operations are very, very different. One operation (sine) finds the ratio of these two sides; the other operation, sine inverse, actually calculates the measure of the angle (B in the example above) using the opposite side and the hypotenuse.

PracticeProblems

Problem 1
Inverse Sine Problem

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

tan-1(24/18) = 53°
Problem 2
Inverse Sine Picture

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

tan-1(5/12) =23°
Problem 3
Inverse sine diagram

What is the measure of the shaded angle on the left?

sin-1(36/39) = 67°
Problem 4

What is the measure of in the triangle below? Round your answer to the nearest tenth of a degree.

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

Use inverse sohcahtoa to calculate the measure of in the triangle below. Round your answer to the nearest hundredth of a degree.

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°

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