A right triangle showing the hypotenuse, opposite and adjacent sides
SOH CAH TOA is a mnemonic device to remember the trigonometric ratios for right triangles. It helps you recall that Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.
Here's a breakdown:
-
SOH:
Sine = Opposite / Hypotenuse
$\sin{\theta} = \frac{\text{opposite}}{\text{hypotenuse}}$ -
CAH:
Cosine = Adjacent / Hypotenuse
$\cos{\theta} = \frac{\text{adjacent}}{\text{hypotenuse}}$ -
TOA:
Tangent = Opposite / Adjacent
$\tan{\theta} = \frac{\text{opposite}}{\text{adjacent}}$
How to use SOH CAH TOA:
- Identify which angle in the right triangle you're working with.
- Determine which sides are opposite, adjacent, and hypotenuse relative to that angle.
- Choose the appropriate ratio (sine, cosine, or tangent) based on which sides you know or need to find.
- Apply the formula to solve for the unknown value.
Example:
In a right triangle, if angle θ = 30° and the hypotenuse is 10 units, find the length of the opposite side.
Step 1: We need to find the opposite side given the hypotenuse and angle, so we use sine:
$\sin{\theta} = \frac{\text{opposite}}{\text{hypotenuse}}$
Step 2: Substitute the values:
$\sin{30°} = \frac{\text{opposite}}{10}$
Step 3: Solve for the opposite side:
$\text{opposite} = 10 \times \sin{30°} = 10 \times 0.5 = 5$ units
Additional Trigonometric Relationships
These three basic trigonometric functions (sine, cosine, and tangent) can also be expressed in terms of each other:
- $\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}$
- $\sin^2{\theta} + \cos^2{\theta} = 1$ (Pythagorean identity)
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