SOHCAHTOA

SOHCAHTOA is a memory aid for the three trigonometric ratios in a right triangle: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse and Tangent = Opposite ÷ Adjacent. Use it to find a missing side when you know an angle and one side, or a missing angle when you know two sides.

What does SOHCAHTOA stand for?

These ratios only work in right triangles, and θ (theta) is one of the two acute angles. For a given angle, each ratio is always the same number however big the triangle is, because all right triangles with that angle are similar. That idea is how high school geometry defines the trig ratios under the Common Core standards.

Opposite, adjacent and hypotenuse

Before you use SOHCAHTOA, label the three sides. The labels depend on which angle you are working with.

• Hypotenuse: the longest side, opposite the right angle. It never changes.
• Opposite: the side across from the angle θ. It doesn't touch θ at all.
• Adjacent: the side next to θ that is not the hypotenuse.

Switch to the other acute angle and the opposite and adjacent sides swap. The hypotenuse stays the same:

Label the sides in this question from the Trigonometry course, then press Mark.

How to use SOHCAHTOA

1. Label the sides opposite, adjacent and hypotenuse from the angle you know or want.
2. Write down what you have and what you want. Ignore the third side.
3. Pick the ratio that uses exactly those two sides: O and H means SOH, A and H means CAH, O and A means TOA.
4. Substitute and solve. For a side, rearrange. For an angle, use the inverse function: sin⁻¹, cos⁻¹ or tan⁻¹.
5. Check that the answer makes sense: the hypotenuse must be the longest side, and the angles must add to 180°.

Finding a missing side

Example 1: the unknown on top

A right triangle has a hypotenuse of 10 and an angle of 35°. Find the side x opposite the angle.

You have the hypotenuse and want the opposite side, so use SOH:

The opposite side is 5.74 to 3 significant figures.

Example 2: the unknown on the bottom

The side opposite a 48° angle is 11. Find the adjacent side x.

Opposite and adjacent means TOA. This time x ends up on the bottom of the fraction, so multiply both sides by x, then divide by tan 48°:

The adjacent side is 9.90. A quick check: the angle is a little more than 45°, so the opposite side should be a little longer than the adjacent side, and 11 is a bit more than 9.90.

Practice choosing the ratio, the step most mistakes come from.

Finding a missing angle

When you know two sides, find the ratio first, then undo it with the inverse function on your calculator.

Example 3: using sin⁻¹

A right triangle has an opposite side of 7 and a hypotenuse of 16. Find the angle θ.

The angle is 25.9°. The other acute angle is 90° − 25.9° = 64.1°.

The Trigonometry calculator solves the whole triangle and shows each step. The SOHCAHTOA calculator guide walks through more examples you can check in it.

Angle of elevation and angle of depression

Many SOHCAHTOA word problems use these two angles. Both are measured from the horizontal.

• Angle of elevation: the angle you look up through, from the horizontal to something above you.
• Angle of depression: the angle you look down through, from the horizontal to something below you.

Example 4: the height of a tree

You stand 30 m from the base of a tree. The angle of elevation to the top is 40°, and your eyes are 1.5 m above the ground. How tall is the tree?

From your eye, the horizontal distance of 30 m is adjacent to the 40° angle, and the height above eye level is opposite. Opposite and adjacent means TOA:

Add your eye height: 25.17 + 1.5 = 26.7 m to 3 significant figures.

Example 5: a boat seen from a cliff

From the top of a cliff 50 m high, the angle of depression to a boat is 18°. How far is the boat from the foot of the cliff?

The angle of depression from the cliff top equals the angle of elevation from the boat up to the cliff top, because they are alternate angles between parallel horizontal lines. So at the boat, the cliff height of 50 m is opposite an 18° angle and the distance d is adjacent:

The boat is about 154 m from the cliff.

SOH CAH TOA questions

Try these, then check the answers below. Give answers to 3 significant figures.

1. The hypotenuse is 15 and the angle is 42°. Find the opposite side.
2. The hypotenuse is 20 and the angle is 65°. Find the adjacent side.
3. The opposite side is 6 and the angle is 30°. Find the adjacent side.
4. The opposite side is 7 and the adjacent side is 10. Find the angle.
5. The adjacent side is 4 and the hypotenuse is 9. Find the angle.
6. The opposite side is 8 and the angle is 25°. Find the hypotenuse.

Answers: 1) 15 sin 42° = 10.0. 2) 20 cos 65° = 8.45. 3) 6 ÷ tan 30° = 10.4. 4) tan⁻¹(7/10) = 35.0°. 5) cos⁻¹(4/9) = 63.6°. 6) 8 ÷ sin 25° = 18.9.

Common mistakes

• Labeling from the wrong angle. Opposite and adjacent depend on which angle you use. Mark θ first.
• Using SOHCAHTOA on a triangle with no right angle. Use the law of sines or the law of cosines instead.
• Calculator in radian mode. Switch to degrees for degree questions.
• Multiplying when the unknown is on the bottom. If x is in the denominator, divide: x = 11 ÷ tan 48°, not 11 × tan 48°.
• Using sin instead of sin⁻¹ to find an angle.
• Measuring elevation or depression from the vertical. Both angles are measured from the horizontal.

Practice with a short timed paper

Right triangle trigonometry comes up across high school geometry, Algebra 2 and the SAT. The daily Geometry, Algebra 2 & SAT Math paper is a short timed practice test of 8 questions, with a new one every day. In the UK, the same practice opens as GCSE Maths Higher.

Read next

Maths · TrigonometrySOHCAHTOA calculator: find a missing side or angleFree SOHCAHTOA calculator: find a missing side or angle of a right triangle with sin, cos or tan, learn which ratio to pick, and check worked examples.