SOHCAHTOA Calculator

How to Calculate Using SOHCAHTOA

SOHCAHTOA is a mnemonic device used to remember the trigonometric ratios in a right triangle. It stands for:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

SOHCAHTOA Formulas

For an angle θ in a right triangle:

1. \( \sin \theta = \frac{opposite}{hypotenuse} \)
2. \( \cos \theta = \frac{adjacent}{hypotenuse} \)
3. \( \tan \theta = \frac{opposite}{adjacent} \)

Calculation Steps

1. Identify the known values (side lengths or angles).
2. Determine which trigonometric ratio to use based on the known and unknown values.
3. Apply the appropriate SOHCAHTOA formula.
4. Solve the equation for the unknown value.
5. Use the Pythagorean theorem to find any remaining side lengths.
6. Calculate other properties such as area and perimeter if needed.

Example Calculation

Let's solve a right triangle with side a = 4 units and angle A = 30°.

1. Given: a = 4 units, A = 30°
2. We can use the sine ratio (SOH) to find the hypotenuse c:

   \( \sin 30° = \frac{4}{c} \)

   \( c = \frac{4}{\sin 30°} = \frac{4}{0.5} = 8 \) units

3. Use the Pythagorean theorem to find side b:

   \( b^2 = c^2 - a^2 = 8^2 - 4^2 = 64 - 16 = 48 \)

   \( b = \sqrt{48} \approx 6.93 \) units

4. Calculate angle B:

   \( B = 90° - A = 90° - 30° = 60° \)

5. Calculate area:

   \( Area = \frac{1}{2} \times a \times b = \frac{1}{2} \times 4 \times 6.93 \approx 13.86 \) square units

6. Calculate perimeter:

   \( Perimeter = a + b + c = 4 + 6.93 + 8 \approx 18.93 \) units

Visual Representation

This diagram illustrates the right triangle from our example, showing how SOHCAHTOA and the Pythagorean theorem are used to solve for all sides and angles.