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. This idea dates back over two thousand years to Greek and Indian astronomers, who needed exactly this kind of angle-to-ratio relationship to calculate the positions of stars and planets.
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 rather than something that has to be recalculated for every differently-sized triangle. That's also why calculators and lookup tables can list sin, cos, and tan for any angle without needing to know anything about a specific triangle at all.
Beyond right-triangle problems, these same three ratios extend to describe periodic motion generally — a rotating point, a sound wave, an alternating electrical current — once the angle θ is allowed to represent something other than a triangle's interior angle, like time or phase. That extension is what the unit circle definition of sine and cosine (used once angles go past 90°) is really doing.
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 →