How to Use SOHCAHTOA

Introduction

What is SOHCAHTOA?

• The term “SOHCAHTOA” is an simple way to help remember the three main trigonometric ratios: Sine, Cosine and Tangent.

What are SOHCAHTOA Rules?

• SOH:

• CAH:

• TOA:

• The side opposite the angle is called the opposite.

• The side touching the angle is called the adjacent.

• The largest side, opposite the right angle, is called the hypotenuse.

Does SOHCAHTOA Only Work for Right Triangles?

How to Use SOHCAHTOA – To Find an Unknown Side of a Right Triangle

1

Label the Sides Clearly

• Identify the triangle’s opposite, adjacent, and hypotenuse sides based on your reference angle.
• Once the sides are labeled, you will have two sides identified—one is given, and the other needs to be found.

2

Use the Correct Trigonometric Ratio

• Based on the given side and the side you need to find, use the correct trigonometric ratio. For example,
• Use Sine Theta (SOH) if you are dealing with opposite and hypotenuse.

• Use Cosine Theta (CAH) if you are dealing with adjacent and hypotenuse.

• Use Tangent Theta (TOA) if you are dealing with opposite and adjacent sides.

3

Solve the Example Briefly

• Use the selected trigonometric formula, substitute the known values, and solve for the missing side with a quick calculation. Here’s a clear example:
• If you know the opposite side and need the hypotenuse, use the Sine formula

Solved Example

A right-angled triangle has a hypotenuse of $10$ cm and an angle of $30^\circ$. Find the length of the opposite side.

SOLUTION

1

Identify the given values:

• Hypotenuse = $10$ cm
• Angle = $30^\circ$

2

Choose the correct trigonometric ratio: Since we are dealing with the opposite side and the hypotenuse, we use Sine Theta.

3

Set up the equation:

$$\sin(30^\circ) = \frac{\text{Opposite}}{10}$$

4

Rearrange the equation to solve for the opposite side:

$$\text{Opposite} = 10 \cdot \sin(30^\circ)$$

5

Substitute the value of $\sin(30^\circ) = 0.5$:

$$\text{Opposite} = 10 \cdot 0.5$$

6

Calculate the final answer:

$$\text{Opposite} = 5$$

Final Answer: $5$ cm

How to Find a Missing Angle of a Right Triangle

1

Identify the Given Sides

• Determine which two sides are provided—opposite, adjacent, or hypotenuse.
• Label the given sides clearly.

2

Use the Correct Trigonometric Ratio

• Use Sine Theta (SOH) if you are dealing with opposite and hypotenuse.

• Use Cosine Theta (CAH) if you are dealing with adjacent and hypotenuse.

• Use Tangent Theta (TOA) if you are dealing with opposite and adjacent sides.

3

Solve for the Angle

• Use the inverse trigonometric function ($\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$) on your calculator to find the angle.
• Rearrange the equation if necessary.
• Calculate to determine the missing angle.

Solved Example

A right-angled triangle has an opposite side of $4$ cm and a hypotenuse of $8$ cm. Find the missing angle.

SOLUTION

1

Identify the given values:

• Opposite = $4$ cm
• Hypotenuse = $8$ cm

2

Choose the correct trigonometric ratio: Since we have the opposite side and the hypotenuse, we use Sine Theta.

3

Set up the equation:

$$\sin(\theta) = \frac{4}{8}$$

4

Simplify the fraction:

$$\sin(\theta) = 0.5$$

5

Use the inverse sine function to find the angle:

$$\theta = \sin^{-1}(0.5)$$

6

Calculate the final answer:

$$\theta = 30^\circ$$

Final Answer: $30^\circ$

Three Additional Solved Examples

Solved Example

A right-angled triangle has an angle of $35^\circ$ and a hypotenuse of $15$ cm. Find the length of the opposite side.

SOLUTION

1

Identify the given values:

• Hypotenuse = $15$ cm
• Angle = $35^\circ$

2

Choose the correct trigonometric ratio: Since we are dealing with the opposite side and the hypotenuse, we use Sine Theta.

3

Set up the equation:

$$\sin(35^\circ) = \frac{\text{Opposite}}{15}$$

4

Rearrange the equation to solve for the opposite side:

$$\text{Opposite} = 15 \cdot \sin(35^\circ)$$

5

Calculate the value:

$$\text{Opposite} = 8.60$$

Final Answer: $8.60$ cm

Solved Example

A right-angled triangle has an adjacent side of $5$ cm and a hypotenuse of $13$ cm. Find the missing angle.

SOLUTION

1

Identify the given values:

• Adjacent = $5$ cm
• Hypotenuse = $13$ cm

2

Choose the correct trigonometric ratio: Since we have the adjacent side and the hypotenuse, we use Cosine Theta.

3

Set up the equation:

$$\cos(\theta) = \frac{5}{13}$$

4

Use the inverse cosine function:

$$\theta = \cos^{-1}\left(\frac{5}{13}\right)$$

5

Calculate the value:

$$\theta = 67.38^\circ$$

Final Answer: $67.38^\circ$

Solved Example

A ladder leans against a wall, reaching a height of $15$ meters. The ladder makes an angle of $65^\circ$ with the ground. Find the length of the ladder.

SOLUTION

1

Identify the given values:

• Opposite side (height) = $15$ m
• Angle = $65^\circ$

2

Choose the correct trigonometric ratio: Since we have the opposite side and the hypotenuse, we use Sine Theta.

3

Set up the equation:

$$\sin(65^\circ) = \frac{15}{\text{Hypotenuse}}$$

4

Rearrange the equation to solve for the hypotenuse:

$$\text{Hypotenuse} = \frac{15}{\sin(65^\circ)}$$

5

Calculate the value:

$$\text{Hypotenuse} = \frac{15}{0.9063}$$

$$\text{Hypotenuse} = 16.55 \text{ m}$$

Final Answer: $16.55$ m