📖Definition
Trigonometry studies the relationships between the angles and side lengths of triangles. In a right triangle, the three basic ratios — sine, cosine, and tangent — connect an angle to the lengths of the triangle's sides.
📐Formula
These are remembered with the mnemonic SOH-CAH-TOA. The hypotenuse is always the side opposite the right angle, while "opposite" and "adjacent" are named relative to whichever angle you are working with.
📝Step-by-Step Guide
Label the sides relative to the angle
Identify the hypotenuse, then decide which remaining side is opposite the angle you know, and which is adjacent to it.
Choose the correct ratio
Pick sine, cosine, or tangent based on which two sides you know or need to find.
Set up the equation
Write the ratio using the known angle and side, with the unknown side as a variable.
Solve for the unknown side or angle
Rearrange and solve, using a calculator in degree mode for the trigonometric value.
🧭SOH-CAH-TOA explained
SOH-CAH-TOA is a mnemonic for the three basic ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Which side counts as "opposite" or "adjacent" always depends on which angle you are measuring from, so relabel the sides each time you switch which angle you are using.
🧭Finding an angle instead of a side
If you know two sides but need the angle, use the inverse trigonometric functions: sin⁻¹, cos⁻¹, or tan⁻¹, sometimes labeled arcsin, arccos, and arctan on a calculator.
For example, if the opposite side is 5 and the hypotenuse is 13, the angle is sin⁻¹(5/13), which is about 22.6°.
⚠️Common Mistakes to Avoid
- Mixing up which side is opposite and which is adjacent for a given angle
- Leaving the calculator in radian mode when the problem uses degrees
- Forgetting that the hypotenuse is always the longest side, opposite the right angle
- Using the wrong ratio for the two sides that are known
- Rounding intermediate steps too early, which compounds errors in the final answer
✏️Practice Problems
In a right triangle, the hypotenuse is 10 and one angle is 30°. Find the length of the side opposite that angle.
💡 Hint: Use sin(30°) = opposite / 10.
Answer: 5 (since sin 30° = 0.5)
In the same triangle, find the side adjacent to the 30° angle.
💡 Hint: Use cos(30°) = adjacent / 10.
Answer: about 8.66
A right triangle has an opposite side of 6 and an adjacent side of 8 relative to angle θ. Find tan(θ).
💡 Hint: tan(θ) = opposite / adjacent.
Answer: 0.75
A ladder leans against a wall, making a 60° angle with the ground. If the ladder is 12 feet long, how high up the wall does it reach?
💡 Hint: The height is opposite the 60° angle, and the ladder is the hypotenuse.
Answer: about 10.39 feet (12 × sin 60°)
Find the angle θ in a right triangle where the opposite side is 5 and the hypotenuse is 13.
💡 Hint: Use the inverse sine function, sin⁻¹(5/13).
Answer: about 22.6°
Printable Trigonometry practice
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❓Trigonometry FAQ
Does trigonometry only work on right triangles?
The basic SOH-CAH-TOA ratios only apply to right triangles. For other triangles, more advanced tools like the Law of Sines and Law of Cosines are used instead.
Should my calculator be in degree or radian mode?
Check what the question uses. Most introductory courses work in degrees, so make sure your calculator is set to degree mode before evaluating sin, cos, or tan.
How do I remember SOH-CAH-TOA?
Many students use the phrase "Some Old Hikers Cant Always Hike To Old Age" or a similar sentence where the first letters spell out SOH-CAH-TOA.