SOH CAH TOA

Trigonometry - Solve Equations - How it Works - Video

Definitions

Definitions:

Here we have the definitions of sine, cosine, and tangent.

The sine of any angle in a right triangle is the opposite side length over the hypotenuse.

The cosine of any angle in a right triangle is the adjacent side length over the hypotenuse.

The tangent of any angle in a right triangle is the opposite side length over the adjacent side length.

Reciprocal Identities

Reciprocal Identities:

Here we have the reciprocal identities.

Example 1

Example 1:

The first step is write down what we know. We are given θ, theta, in the top left of the triangle. To make things less complicated, we are going to keep using this angle for all the trigonometric functions.

We can use SOH CAH TOA to write down what we need to find.

Example 1 continued:

Here we have our last step. Since θ, theta, is in the top left, the opposite side length is the orangish side, 3; the adjacent side is the blue side length, 4; the hypotenuse is the green side length, 5.

Then we substitute what we know into what we have so ...

30 - 60 - 90 Triangle

30 - 60 - 90 Triangle:

You will be ask to write exact answers, and if you have a triangle that has angles of 30°, 60°, and 90°, there are exact measurements. It will be easier to memorize this measurements, but not impossible if you don't. It will just take longer.

We start out with an equilateral triangle with sides of 2 and cut it in half. Now we have two triangles with angles of 30°, 60°, and 90° with a height of sqrt(3). We can use the Pythagorean theorem to find that length. Can we use any equilateral triangle? What happens if we use a bigger one?

It does matter whether we focus on the triangle on the left or the triangle on the right. It will give us sin(30°) = 1/2, cos(30°) = sqrt(3) / 2, and tan(30°) = sqrt(3) / 3 and sin(60°) = sqrt(3) /2, cos(60°) = 1 / 2, and tan(60°) = sqrt(3)

45 - 45 - 90 Triangle

45 - 45 - 90 Triangle:

You will be ask to write exact answers, and if you have a triangle that has angles of 45°, 45°, and 90°, there are exact measurements. It will be easier to memorize this measurements, but not impossible if you don't. It will just take longer.

Can we use any isosceles triangle? What happens if we use a bigger one?

It does matter whether we focus on the vertical or horizontal side length of the right triangle. It will give us sin(45°) = sqrt(2) /2, cos(45°) = sqrt(2) / 2, and tan(45°) = 1.

Example 2

Example 2:

Now that we know the special triangles of 30 - 60 - 90 and 45 - 45 - 90 we can find some exact measurements.

The first step is write down what we know. We are given θ, 60°, in the top left of the triangle. To make things less complicated, we are going to keep using this angle for all the trigonometric functions.

We can use SOH CAH TOA to write down what we need to find.

Example 2 continued:

Here we have our last step. Since θ, 60°, is in the top left, the opposite side length is the orangish side, m; the adjacent side is the blue side length, 5; the hypotenuse is the green side length, n.

Then we substitute what we know into what we have so ...

Example 2 continued:

We have found our answers: m, 5 * sqrt(3), and n, 10. Let's continue to find sin(60°).

Example 2 continued:

That is same as the model above so we did it correct.

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