# TRIGONOMETRIC RATIOS & SOH CAH TOA

In this section we learn about the trigonometric ratios as well as SOH CAH TOA, which is an acronym for memorizing the ratios.

Each of the three trigonometric ratios is listed below. In each case we state the formula as well as illustrate it with two examples (one for each of the interior angles, $$a$$ and $$b$$, of the triangle). Make a note of each of the ratios and make sure to notice that the value of sine, cosine, tangent, depends on the interior angle that we're focusing on; we calculat the sine, cosine and tangent of an angle not of a triangle.

### SINE = Opposite $$\div$$ Hypotenuse

Given a right angle triangle and one of its interior angles $$a$$, the sine of angle $$a$$ equals to:

its opposite side length divided by the length of the hypotenuse.
That's: $sin(a) = \frac{\text{Opposite Side Length}}{\text{Hypotenuse}}$ Rather than writing "Opposite Side Length" and "Hypotenuse", we usually just use the letters O and H: $sin(a) = \frac{\text{O}}{\text{H}}$

To see some examples click on the button show here:

#### Examples

Remembering that $$sin(a) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{H}$$, we calculate the sine of the two (acute) angles, $$a$$ and $$b$$, of the right angle triangle shown. We do this relative to each of the two interior (acute) angles $$a$$ and $$b$$.

#### Relative to angle $$a$$

We calculate the sine of angle $$a$$, $$sin(a)$$, as follows:

\begin{aligned} sin(a) &= \frac{O}{H} \\ & = \frac{3}{5} \\ sin(a) & = 0.6 \end{aligned}

#### Relative to angle $$b$$

We calculate the sine of angle $$b$$, $$sin(b)$$, as follows:

\begin{aligned} sin(b) &= \frac{O}{H} \\ & = \frac{4}{5} \\ sin(b) & = 0.8 \end{aligned}

Notice that depending on the angle we're looking at, $$a$$ or $$b$$, the Opposite Side Length changes. Consequently the value of sine changes depending on whether we're cosidering the angle $$a$$ or the angle $$b$$.

### COSINE = Adjacent $$\div$$ Hypotenuse

Given a right angle triangle and one of its interior angles $$a$$, the sine of angle $$a$$ equals to:

its adjacent side length divided by the length of the hypotenuse.
That's: $cos(a) = \frac{\text{Adjacent Side Length}}{\text{Hypotenuse}}$ Rather than writing "Adjacent Side Length" and "Hypotenuse", we usually just use the letters A and H: $cos(a) = \frac{\text{A}}{\text{H}}$

To see some examples click on the button show here:

#### Examples

Remembering that $$cos(a) = \frac{\text{Adjacent}}{\text{Adjacent}} = \frac{A}{H}$$, we calculate the cosine of the two (acute) angles, $$a$$ and $$b$$, of the right angle triangle shown. We do this relative to each of the two interior (acute) angles $$a$$ and $$b$$.

#### Relative to angle $$a$$

We calculate the sine of angle $$a$$, $$cos(a)$$, as follows:

\begin{aligned} cos(a) &= \frac{A}{H} \\ & = \frac{4}{5} \\ cos(a) & = 0.8 \end{aligned}

#### Relative to angle $$b$$

We calculate the sine of angle $$b$$, $$cos(b)$$, as follows:

\begin{aligned} cos(b) &= \frac{O}{H} \\ & = \frac{3}{5} \\ cos(b) & = 0.6 \end{aligned}

Notice that depending on the angle we're looking at, $$a$$ or $$b$$, the Adjacent Side Length changes. Consequently the value of cosine changes depending on whether we're cosidering the angle $$a$$ or the angle $$b$$.

### TANGENT = Opposite $$\div$$ Adjacent

Given a right angle triangle and one of its interior angles $$a$$, the tangent of angle $$a$$, written $$tan(a)$$, equals to:

its opposite side length divided by its adjacent side length.
That's: $tan(a) = \frac{\text{Opposite Side Length}}{\text{Adjacent Side Length}}$ Rather than writing "Opposite Side Length" and "Adjacent Side Length", we usually just use the letters O and A: $tan(a) = \frac{\text{O}}{\text{A}}$

To see some examples click on the button show here:

#### Examples

Remembering that $$tan(a) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}$$, we calculate the tangent of the two (acute) angles, $$a$$ and $$b$$, of the right angle triangle shown. We do this relative to each of the two interior (acute) angles $$a$$ and $$b$$.

#### Relative to angle $$a$$

We calculate the sine of angle $$a$$, $$tan(a)$$, as follows:

\begin{aligned} tan(a) &= \frac{O}{A} \\ & = \frac{3}{4} \\ tan(a) & = 0.75 \end{aligned}

#### Relative to angle $$b$$

We calculate the sine of angle $$b$$, $$tan(b)$$, as follows:

\begin{aligned} tan(b) &= \frac{O}{A} \\ & = \frac{4}{3} \\ tan(b) & = 1.33... \end{aligned}

Notice that depending on the angle we're looking at, $$a$$ or $$b$$, the Adjacent and the Opposite Side Lengths changes. Consequently the value of tangent changes depending on whether we're cosidering the angle $$a$$ or the angle $$b$$.

## Exercise 1

For each of the right angle triangles, below, calculate:

1. $$sin(a)$$
2. $$cos(a)$$
3. $$tan(a)$$
1. $$sin(b)$$
2. $$cos(b)$$
3. $$tan(b)$$

#### Triangle 4

Note: this exercise can be downloaded as a worksheet to practice with:

## Solution Without Working

#### Triangle 1

1. $$sin(a) = 0.882$$
2. $$cos(a) = 0.471$$
3. $$tan(a) = 1.88$$

1. $$sin(b) = 0.471$$
2. $$cos(b) = 0.882$$
3. $$tan(b) = 0.533$$

#### Triangle 2

1. $$sin(a) = 0.6$$
2. $$cos(a) = 0.8$$
3. $$tan(a) = 0.75$$

1. $$sin(b) = 0.8$$
2. $$cos(b) = 0.6$$
3. $$tan(b) = 1.33$$

#### Triangle 3

1. $$sin(a) = 0.923$$
2. $$cos(a) = 0.385$$
3. $$tan(a) = 2.4$$

1. $$sin(b) = 0.385$$
2. $$cos(b) = 0.923$$
3. $$tan(b) = 0.417$$

#### Triangle 4

1. $$sin(a) = 0.690$$
2. $$cos(a) = 0.724$$
3. $$tan(a) = 0.952$$

1. $$sin(b) = 0.724$$
2. $$cos(b) = 0.690$$
3. $$tan(b) = 1.05$$

## SOH CAH TOA

SOH CAH TOA is an acronym used to memorize the three trigonometric ratios.

Indeed simply say "SOH CAH TOA" a few times and you'll remember those "words".

Here's what they stand for:

### Tutorial: Labelling Sides in Right Angle Triangles

In the following tutorial we learn how to label the sides in a right angle triangle.

## SOH

SOH gives us the first letter of each of words we need to know to remember the trigonometric ratio for the sine of an angle $$a$$:

Which quickly leads us to: $sin(a)=\frac{O}{H}$

## CAH

CAH gives us the first letter of each of words we need to know to remember the trigonometric ratio for the cosine of an angle $$a$$:

Which quickly leads us to: $cos(a)=\frac{A}{H}$

## TOA

TOA gives us the first letter of each of words we need to know to remember the trigonometric ratio for the tangent of an angle $$a$$:

Which quickly leads us to: $tan(a)=\frac{O}{A}$