SOHCAHTOA is a mnemonic device that helps you remember the three primary trigonometric ratios. Each group of three letters represents one ratio:
SOH
Sine = Opposite / Hypotenuse
sin θ = O / H
CAH
Cosine = Adjacent / Hypotenuse
cos θ = A / H
TOA
Tangent = Opposite / Adjacent
tan θ = O / A
These three ratios are the core of right-triangle trigonometry. Every problem involving a right triangle and an angle can be solved using one (or more) of these ratios. The key is knowing which ratio to use — and that starts with correctly identifying the sides.
Identifying Opposite, Adjacent, and Hypotenuse
Before you can apply any trig ratio, you must correctly label the three sides of the right triangle relative to the angle you are working with. This is a crucial point that students often miss: the labels "opposite" and "adjacent" change depending on which angle you choose.
Hypotenuse: Always the longest side, directly opposite the right angle. This never changes regardless of which acute angle you pick.
Opposite: The side directly across from the angle θ you are working with. It does not touch the angle.
Adjacent: The side that touches the angle θ but is not the hypotenuse. It forms one of the rays of the angle.
Key Takeaway
The hypotenuse is always the same, but "opposite" and "adjacent" depend on which angle you pick. Always ask: "Opposite to WHICH angle?" before writing any trig ratio.
Step-by-Step Method for Solving Right Triangle Problems
Here is a systematic approach that works for every SOHCAHTOA problem:
Draw and label the triangle. Mark the right angle, the known angle (or the angle you need to find), and all known side lengths.
Identify the sides. Label the opposite, adjacent, and hypotenuse relative to the angle you are working with.
Choose the right ratio. Look at what you know and what you need:
If you have (or need) the opposite and hypotenuse, use sine (SOH)
If you have (or need) the adjacent and hypotenuse, use cosine (CAH)
If you have (or need) the opposite and adjacent, use tangent (TOA)
Set up the equation. Write the trig ratio with the known and unknown values.
Solve. Use algebra to isolate the unknown. If finding a side, multiply or divide. If finding an angle, use the inverse trig function.
Finding a Missing Side
When you know one acute angle and one side of a right triangle, you can find any other side using SOHCAHTOA. The process involves choosing the correct ratio, substituting your known values, and solving for the unknown side.
To find the opposite: O = H × sin θ | O = A × tan θ
To find the adjacent: A = H × cos θ | A = O / tan θ
To find the hypotenuse: H = O / sin θ | H = A / cos θ
Finding a Missing Angle
When you know two sides of a right triangle but need to find an angle, you use the inverse trig functions (also called arc functions). These "undo" the trig ratio to give you the angle:
On most calculators, these functions are labeled sin−1, cos−1, and tan−1. Make sure your calculator is set to degrees mode (unless you need radians).
Choosing the Right Ratio
A common source of confusion is knowing which of the three ratios to use. Here is a quick decision guide:
Look at the angle you are working with.
Identify which two of the three sides (opposite, adjacent, hypotenuse) are involved in your problem — one will be known, one will be unknown.
Pick the ratio that connects those two sides:
Opposite + Hypotenuse → Sine
Adjacent + Hypotenuse → Cosine
Opposite + Adjacent → Tangent
If neither of the two sides you have matches a single ratio, you may need to use the Pythagorean theorem first to find a third side, then apply SOHCAHTOA.
Common Mistakes to Avoid
Even after learning the ratios, students frequently make these errors:
Confusing opposite and adjacent: Remember that these labels are relative to the angle you choose, not fixed to the triangle. Always re-identify them for each problem.
Calculator in wrong mode: If your calculator is set to radians when you expect degrees (or vice versa), your answers will be wildly off. Always check your mode.
Dividing when you should multiply (and vice versa): If sin(θ) = O/H, then O = H × sin(θ), not O = sin(θ) / H. Be careful with your algebra.
Using SOHCAHTOA on non-right triangles: These ratios only work for right triangles. For other triangles, use the Law of Sines or Law of Cosines.
Forgetting to check reasonableness: A side length should never be negative. An angle in a triangle should be between 0° and 90° (for the acute angles). Always sanity-check your answer.
Worked Examples
Example 1: Find a Missing Side
In a right triangle, angle A = 35° and the hypotenuse is 20 cm. Find the length of the side opposite angle A.
Step 1: We have the angle (35°) and the hypotenuse (20 cm). We need the opposite side.
Step 2: The ratio connecting opposite and hypotenuse is sine (SOH).
Step 3: Set up the equation: sin(35°) = O / 20
Step 4: Solve for O: O = 20 × sin(35°) = 20 × 0.5736 = 11.47 cm
Answer: The opposite side is approximately 11.47 cm.
Example 2: Find a Missing Angle
In a right triangle, the side opposite angle B is 7 m and the adjacent side is 10 m. Find angle B.
Step 1: We have the opposite (7 m) and adjacent (10 m). We need the angle.
Step 2: The ratio connecting opposite and adjacent is tangent (TOA).
Step 3: Set up: tan(B) = 7 / 10 = 0.7
Step 4: Use inverse tangent: B = tan¹(0.7) = 34.99°
Answer: Angle B is approximately 35.0°.
Example 3: Real-World Problem — Ladder Against a Wall
A 6-meter ladder leans against a wall, making a 72° angle with the ground. How high up the wall does the ladder reach?
Step 1: The ladder is the hypotenuse (6 m). The angle with the ground is 72°. The wall height is opposite the ground angle.
Step 2: We need the opposite side (wall height) and have the hypotenuse. Use sine.
Answer: The ladder reaches approximately 5.71 meters up the wall.
Practice Problems
Practice 1: Find the Hypotenuse
In a right triangle, angle C = 28° and the side adjacent to angle C is 15 cm. Find the hypotenuse.
We have the adjacent side and need the hypotenuse, so use cosine: cos(28°) = 15 / H. Therefore H = 15 / cos(28°) = 15 / 0.8829 = 16.99 cm (approximately 17.0 cm).
Practice 2: Find an Angle
A right triangle has a hypotenuse of 25 and a side of length 7 opposite the unknown angle θ. Find θ.
We have the opposite and hypotenuse, so use sine: sin(θ) = 7 / 25 = 0.28. Therefore θ = sin¹(0.28) = 16.26° (approximately 16.3°).
Practice 3: Multi-Step Problem
From the top of a 50-meter cliff, the angle of depression to a boat is 40°. How far is the boat from the base of the cliff?
The angle of depression from the top equals the angle of elevation from the boat (alternate interior angles), so the angle at the boat is 40°. The cliff height (50 m) is opposite the 40° angle, and the horizontal distance is adjacent. Use tangent: tan(40°) = 50 / d. Therefore d = 50 / tan(40°) = 50 / 0.8391 = 59.59 meters (approximately 59.6 m).
Key Takeaway
SOHCAHTOA gives you three powerful tools. For every right triangle problem: (1) label the sides relative to your angle, (2) pick the ratio that connects the two sides you care about, and (3) solve. Master this process and you can handle any right-triangle trig problem.
Frequently Asked Questions
When do I use sine vs cosine vs tangent?
Use sine when your problem involves the opposite side and the hypotenuse. Use cosine when it involves the adjacent side and the hypotenuse. Use tangent when it involves the opposite and adjacent sides. The trick is to identify which two sides are relevant to your problem and pick the ratio that connects them.
What if the angle is not given in the problem?
If you know two sides but not the angle, you can find the angle using inverse trig functions. For example, if you know the opposite and adjacent sides, use θ = tan¹(opposite / adjacent). If you know no angles at all (only all three sides), you can use the inverse trig functions with any pair of sides to find an angle.
Does SOHCAHTOA work for all triangles?
No, SOHCAHTOA only works for right triangles. For non-right triangles, you need the Law of Sines (a/sin A = b/sin B = c/sin C) or the Law of Cosines (c² = a² + b² − 2ab cos C). These more general formulas reduce to SOHCAHTOA when the triangle happens to be a right triangle.
What does SOHCAHTOA stand for?
SOHCAHTOA stands for: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. Some students remember it with the phrase "Some Old Hippie Caught Another Hippie Tripping On Acid," though there are many creative variations.
How do I remember which side is opposite and which is adjacent?
Think of it relative to the angle you are working with. The opposite side is directly across the triangle from your angle — it does not touch the angle at all. The adjacent side touches the angle but is not the hypotenuse. The hypotenuse is always across from the 90° angle and is the longest side.
Can I use SOHCAHTOA to find an angle of elevation or depression?
Yes, angles of elevation and depression problems are classic applications of SOHCAHTOA. The angle of elevation is measured upward from horizontal, and the angle of depression is measured downward from horizontal. In both cases, you form a right triangle with a horizontal base, a vertical height, and a line of sight, then apply the appropriate trig ratio.