# SohCahToa: Definition & Example Problems

## Trigonometric Functions

Sine, cosine, and tangent are the three main functions in trigonometry. They're all based on ratios obtained from a right triangle. Before we can discuss what ratios work for which function, we need to label a right triangle.

**Opposite** is the side opposite the angle in question, **adjacent** is the side next to the angle in question, and the **hypotenuse** is the longest side of a right triangle. The hypotenuse is always opposite the right angle.

The ratios that allow you to determine the sine, cosine, and tangent of a right triangle are:

- The
**sine**of an angle is equal to the side opposite the angle divided by the hypotenuse. - The
**cosine**of an angle is equal to the side adjacent to the angle divided by the hypotenuse. - The
**tangent**of an angle is equal to the side opposite the angle divided by the side adjacent to the angle.

## SOHCAHTOA

These ratios can be difficult to remember. You might easily get confused and not remember which side goes where. **SOHCAHTOA** is a mnemonic device helpful for remembering what ratio goes with which function.

- SOH =
**S**ine is**O**pposite over**H**ypotenuse - CAH =
**C**osine is**A**djacent over**H**ypotenuse - TOA =
**T**angent is**O**pposite over**A**djacent

With these properties, you can solve almost any problem related to finding either a side length or angle measure of a right triangle. SohCahToa can ensure that you won't get them wrong.

## Examples

Now, for some example problems. Let's find *x* for this triangle.

We know the side adjacent to the known angle of 60 degrees is 13 cm. We're trying to find the length of the side opposite the known angle of 60 degrees. Thus, we need to use TOA, or tangent (tan), which uses opposite and adjacent. Therefore, our equation will be:

tan 60 = *x*/13

The tan of 60 is 1.73, which makes the equation:

1.73 = *x*/13

Solve for *x* to get:

*x* = (1.73) * (13) = 22.49

So, the length of side *x* is 22.49 cm.

On to another example problem. What is the sine of 35 degrees?

The sine of an angle is equal to the opposite side divided by the hypotenuse.

sin 35 = 2.8 / 4.9 = 0.57

And for one final problem, find sin, cos and tan for the angle in the right triangle shown.

sin = opposite/hypotenuse = 3 / 5 = 0.6

cos = adjacent/hypotenuse = 4 / 5 = 0.8

tan = opposite/adjacent = 3 / 4 = 0.75

## Real-Life Examples

Trigonometric functions are important for many reasons. They let you calculate angles when you know sides and calculate sides when you know angles. This can be helpful in many real life situations, such as when determining the height of a large building or the distance across a lake - things that just can't be measured easily.

For instance, let's imagine a boat is anchored in the ocean. The rope attached to the anchor is 30 meters long. The angle that the anchor makes with the ocean floor is 39 degrees. What is the depth of the ocean?

Since it isn't practical to dive down and measure how deep the anchor is, we can use a trigonometric ratio to figure it out. We know the angle the cable makes with the ocean floor and the length of the cable (hypotenuse). To find the length of the side opposite the angle *d*, we use the sine function.

sin 39 = *d*/30

0.63 = *d*/30

*d* = 18.9 m

Let's try another one. What is the height of this building?

This time, we know an angle and the side adjacent to that angle, and we're trying to find the side opposite *h*. We need to use the ratio that includes both opposite and adjacent, which is the tangent (TOA).

tan 55 = *h*/57

1.43 = *h*/57

*h* = 81.4 ft

## Lesson Summary

The trigonometric ratios **sine**, **cosine**, and **tangent** are all calculated using ratios from a right triangle. The mnemonic **SOHCAHTOA** can be used to aid in remembering which function to use in what circumstance - **SOH** stands for Sine is opposite over hypotenuse; **CAH** stands for Cosine is adjacent over hypotenuse; and **TOA** stands for Tangent is opposite over adjacent. This will save confusion when working with these functions.

## Lesson at a Glance

SOHCAHTOA helps us remember which trigonometric function to use to calculate ratios from a right triangle and find either a side length or angle measure. This can be useful in real-world applications, such as finding the depth of the ocean or the height of a skyscraper.

- SOH = Sine is Opposite over Hypotenuse
- CAH = Cosine is Adjacent over Hypotenuse
- TOA = Tangent is Opposite over Adjacent

## Learning Outcomes

When you are done reading about SOHCAHTOA, you should be able to

- Define sine, cosine, and tangent
- Describe what SOHCAHTOA is and how it can be applied in trigonometric functions
- Calculate lengths and ratios using SOHCAHTOA

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## Practice Questions

1. For the triangle ABC shown below, use *SohCahToa* to find side AB, side BC. Then write down the measure of angle B.

2. A rope is attached to the top of a vertical, straight tree. The other end of the rope is tied to a point on the level ground at a distance of 20 feet from the base of the tree which is also on level ground. The rope makes an angle of 60 degrees with the level ground. Use *SohCahToa* to calculate the height of the tree.

## Answers

1. Using *Toa* we define

tan 35 = Opposite/Adjacent = AB/AC = AB/12.

Then cross multiplying the above relation gives us

Next, using *Cah* we get

cos 35 = Adjacent/Hypotenuse = AC/BC = 12/BC.

Cross multiplying the above relation leads to

Finally, since ABC is a right triangle with angle A = 90 degrees and angle C = 35 degrees, angle B will equal 180 - (90 + 35) = 180 - 125 = 55 degrees.

**Hence, AB = 8.40 units, BC = 14.65 units and angle B = 55 degrees.**

2. Let the top of the rope attached to the top of the tree be point A. Then let B be the base of the tree. Finally let C be the point where the rope is attached to the level ground.

Since AB is perpendicular to BC, therefore ABC is a right triangle.

Next, BC = 20 feet and angle C = 60 degrees.

Hence to find the height of the tree or AB, we use *Toa* in the triangle ABC to get

tan C = Opposite/Adjacent = AB/BC

OR

tan 60 = AB/20.

Cross multiplying the above relation yields

**The height of the tree is 34.64 feet.**

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