SOHCAHTOA

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SOHCAHTOA: meaning, method and solved example (Geometry) – Maths PDF

In short: SOHCAHTOA squeezes the three basic right-triangle ratios into one pronounceable word, with SOH for sine, CAH for cosine and TOA for tangent, each tying an angle to two sides.

GeometryTrigonometryG-SRT.C.6

What is SOHCAHTOA?

SOHCAHTOA is read in three chunks, one per ratio: \(\sin\theta = \frac{\text{opp}}{\text{hyp}}\), \(\cos\theta = \frac{\text{adj}}{\text{hyp}}\) and \(\tan\theta = \frac{\text{opp}}{\text{adj}}\).

Sides are named from a chosen acute angle \(\theta\). Opposite looks at \(\theta\) from across the triangle, adjacent touches \(\theta\), and the hypotenuse is the slanted, longest side facing the right angle. Because each ratio links an angle to a pair of sides, the trio finds a missing side or a missing angle.

How to spot it

  • A right triangle is given with one acute angle and one side, and another side is requested.
  • Two sides are known and the problem asks for an angle in degrees.
  • A ladder, ramp, kite string or wire makes an angle with the ground and a height or distance is needed.
  • The words sine, cosine or tangent appear together with a right-angle mark.

The method

  1. Mark the angle \(\theta\), then label each side as opposite, adjacent or hypotenuse relative to it.
  2. Choose the chunk that contains the two sides involved, one known and one unknown.
  3. Write the equation, such as \(\sin\theta = \frac{\text{opp}}{\text{hyp}}\), and solve for the unknown side.
  4. To find an angle, use the inverse key: \(\theta = \tan^{-1}\left(\frac{\text{opp}}{\text{adj}}\right)\).
θoppositeadjacenthypotenuseSOH: sin = opp / hypCAH: cos = adj / hypTOA: tan = opp / adj
Sides are named from the angle θ: opposite faces it, adjacent touches it, the hypotenuse faces the right angle.

SOHCAHTOA: solved example

Problem

A survey drone hovers 36 meters above a flat field. Its straight-line distance to a landing pad is 60 meters. At what angle \(\theta\) below the horizontal must the camera point at the pad?

Solution

The height 36 m faces \(\theta\), so it is opposite; the 60 m line of sight is the hypotenuse. Those two sides call for the sine chunk: \(\sin\theta = \frac{36}{60} = 0.6\).

Use the inverse key: \(\theta = \sin^{-1}(0.6) \approx 36.87^{\circ}\). The camera points about \(36.9^{\circ}\) below horizontal.

Try it

Question 1

A 30-foot extension ladder leans on a wall at \(70^{\circ}\) to the ground. How high up the wall does it reach? Use \(\sin 70^{\circ} \approx 0.9397\).

Show the answer

The height is opposite and the ladder is the hypotenuse, so \(30 \times 0.9397 \approx 28.2\) feet.

Question 2

A 50-meter support cable makes \(35^{\circ}\) with level ground. How far from the pole does it land? Use \(\cos 35^{\circ} \approx 0.8192\).

Show the answer

The ground distance is adjacent, so \(50 \times 0.8192 \approx 41.0\) meters.

SOHCAHTOA vs. Pythagorean Theorem

The Pythagorean theorem relates three sides to each other, \(a^2 + b^2 = c^2\), and needs no angle. The memory phrase here ties an angle to two sides, so it is the tool when an angle is known or wanted. Two sides given, third side wanted: Pythagorean. One angle involved: SOHCAHTOA.

Pythagorean Theorem

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