Find Measures of Angles Using Sin, Cos, Tan

Using SOH-CAH-TOA to Find Angles

Trigonometric functions (sin, cos, and tan) can be used to find the lengths of the sides of a right triangle. You can review that here.

The same functions can also be used to find the measures of angles in a right triangle.

$$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$$

Based on the formulas we've got above, you can solve for angles ($\theta$) if you have some or all of the lengths of the triangle's sides.

Let's try an example.

To solve for an angle, we need to use the inverse of the trig function we choose. We abbreviate these inverses like this:

$\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$.

This involves a different button on your calculator, so be mindful of this.

(Video) Trigonometric Functions to Find Unknown Sides of a Right Triangle

by PatrickJMT

Here's a video by PatrickJMT showing you how to length of a side of a right triangle given and angle.

$\large{\text{Summary}}$

Sine, cosine, and tangent are all trigonometric functions that can be used to help you find the angles or lengths of a triangle.

When finding the angle of $\theta$ we use the trig functions. $$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\\ \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\\ \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$$

A mnemonic way to remember this is the word SOH-CAH-TOA

SOH stands for Sin is Opposite over Hypotenuse

CAH stands for Cos is Adjacent over Hypotenuse

TOA stands for Tan is Opposite over Adjacent

(Video) Inverse Trigonometric Functions

by PatrickJMT

Here's a video by PatrickJMT solving some inverse trig functions and showing you how to do it.

$\large{\text{Summary}}$

When solving for the angles of a triangle, it's important to know inverse trig functions. The equations for the three basic trig functions are: $$\sin(\theta)=\dfrac{\text{opposite}}{\text{hypotenuse}}\\ \cos(\theta)=\dfrac{\text{adjacent}}{\text{hypotenuse}}\\ \tan(\theta)=\dfrac{\text{opposite}}{\text{adjacent}}$$

(Video) Working out Unknown Angles with Basic Trigonometry

by tecmath

In this video by tecmath it is shown how to find the missing angles of a triangle using

$\large{\text{Summary}}$

Trigonometry is not as scary as it sounds! To solve these problems, you need to be able to label the sides of a right triangle based on the angle theta.

The steps for solving these problems are:

• Label sides
• Work out trig function
• Substitute values
• Solve

(Video) Sin, Cos, Tan

by Nancy Pi

In this video by Nancy Pi the concepts of trigonometric functions (sin, cos, tan) are explained.

$\large{\text{Summary}}$

Here are the trig equations that are important to know: $$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$$ $$\csc(\theta)=\frac{\text{hypotenuse}}{\text{opposite}}$$ $$\sec(\theta)=\frac{\text{hypotenuse}}{\text{adjacent}}$$ $$\cot(\theta)=\frac{\text{adjacent}}{\text{opposite}}$$

If you are given the following information:

$\text{hypotenuse} = 5$ $\\$ $\text{adjacent side} = 3$ $\\$ $\text{opposite side} = 4$ $\\$

In this problem, $$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{4}{5}$$ $$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{3}{5}$$ $$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{4}{3}$$

Now, let's find cosecant, secant, and cotangent. You find these by finding the inverse or partner and take the reciprocal. So to find cosecant, we take $\sin(\theta)=\frac{4}{5}$ and take the reciprocal which is $\frac{1}{\frac{4}{5}}$ which is equal to $\frac{5}{4}$.

We do something similar for secant and cotangent.

(Video) Sin Cos Tan

by Math Meeting

In this video by Math Meeting the trigonometric functions sine, cosine, and tangent are explained.

$\large{\text{Summary}}$

$$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$$

If you are given the following information:

$\text{hypotenuse} = 5$ $\\$ $\text{adjacent side} = 3$ $\\$ $\text{opposite side} = 4$ $\\$

In this problem, $$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{4}{5}$$ $$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{3}{5}$$ $$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{4}{3}$$