Sine, Cosine & Tangent
What is the Sine, Cosine & Tangent?
Sine, cosine, and tangent are the three fundamental trigonometric ratios. For a right triangle with angle θ, they describe fixed relationships between that angle and ratios of the triangle's side lengths — remembered with the mnemonic SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
These ratios only depend on the angle, not the triangle's size — any right triangle with the same angle θ gives the exact same sin θ, cos θ, and tan θ, which is what makes them useful as universal lookup values.
What Each Variable Means
When to Use It
- Finding a missing side length in a right triangle when one side and one angle are known
- Finding a missing angle when two sides are known (using the inverse functions sin⁻¹, cos⁻¹, tan⁻¹)
- As the foundation for the Law of Sines and Law of Cosines, which extend these ratios to any triangle
Step-by-Step Example
Problem: A right triangle has a hypotenuse of 10 and an opposite side of 6. Find the angle θ.
Opposite and Hypotenuse are known, so use sine.
sin θ = O/H = 6/10 = 0.6Undo the sine to solve for the angle itself.
θ = sin⁻¹(0.6)Interactive Calculator
Common Mistakes
Mistake: Mixing up which side is opposite versus adjacent.
Fix: Opposite and adjacent are defined relative to the specific angle θ you're working with — the same side can be "opposite" for one angle and "adjacent" for the other non-right angle in the same triangle.
Mistake: Using degrees and radians inconsistently on a calculator.
Fix: Check your calculator's angle mode before computing — sin(30) gives a very different (and usually wrong) answer in radian mode versus degree mode.
Practice Questions
A right triangle has an adjacent side of 8 and a hypotenuse of 17. Find cos θ.
A right triangle has opposite side 5 and adjacent side 12. Find tan θ.
Frequently Asked Questions
Do SOH-CAH-TOA ratios work for non-right triangles?
No — they're only defined relative to a right angle. For general triangles, use the Law of Sines or Law of Cosines instead.
Why is tan(90°) undefined?
Because tan θ = sin θ / cos θ, and cos(90°) = 0 — dividing by zero is undefined.
Related Formulas
Pythagorean Theorem
Relates the three sides of a right triangle, letting you find any one side from the other two.
Learn more →Law of Sines
Relates the sides of any triangle to the sines of their opposite angles — not just right triangles.
Learn more →Law of Cosines
A generalization of the Pythagorean theorem that works for any triangle, not just right triangles.
Learn more →