Understanding SOHCAHTOA in Trigonometry

What is SOHCAHTOA?

SOHCAHTOA is a mnemonic that helps us remember the definitions of the trigonometric ratios: sine, cosine, and tangent. These ratios are used to find unknown sides or angles in right-angled triangles.

• Sine (SOH): $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine (CAH): $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent (TOA): $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

Here, $\theta$ is the angle in the triangle, and the sides are labeled relative to this angle:

• Opposite: The side opposite the angle $\theta$.
• Adjacent: The side next to the angle $\theta$, not the hypotenuse.
• Hypotenuse: The longest side of the right-angled triangle.

Using SOHCAHTOA

To solve problems using SOHCAHTOA, follow these steps:

1. Identify the sides: Label the sides of the triangle as opposite, adjacent, or hypotenuse based on the angle $\theta$.
2. Select the ratio: Choose the appropriate trigonometric ratio (sine, cosine, or tangent) based on the sides you know and need to find.
3. Set up the equation: Write the equation using the chosen ratio.
4. Solve the equation: Rearrange and solve for the unknown side or angle.

Examples

Example 1: Finding a Side

Example 2: Finding an Angle