Mathwords logoReference LibraryMathwords

Sine vs. Cosine vs. Tangent

Sine, cosine, and tangent are the three basic trigonometric ratios. In a right triangle: sine = opposite ÷ hypotenuse, cosine = adjacent ÷ hypotenuse, tangent = opposite ÷ adjacent. The mnemonic SOHCAHTOA helps you remember. All three are related: tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}.

Sine vs. Cosine

SineCosine
Definition (right triangle)sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
Tangenttan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}
Range[−1,1][-1, 1][−1,1][-1, 1]
Period2π2\pi (360°)2π2\pi (360°)
At 0°sin⁡0°=0\sin 0° = 0cos⁡0°=1\cos 0° = 1
At 90°sin⁡90°=1\sin 90° = 1cos⁡90°=0\cos 90° = 0
Relationshipsin⁡θ=cos⁡(90°−θ)\sin\theta = \cos(90° - \theta)cos⁡θ=sin⁡(90°−θ)\cos\theta = \sin(90° - \theta)
Pythagorean identitysin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

When to Use Each

Use Sine when...

  • Finding a side opposite a known angle in a right triangle
  • Modeling wave motion or periodic phenomena
  • Law of Sines problems (non-right triangles)
  • Finding the vertical component of a vector

Use Cosine when...

  • Finding a side adjacent to a known angle
  • Law of Cosines problems (when you know SAS)
  • Finding the horizontal component of a vector
  • Dot product calculations

Examples

SOHCAHTOA example
A right triangle has a 30° angle, with hypotenuse = 10. The side opposite the 30° angle = 10×sin⁡30°=10×0.5=510 \times \sin 30° = 10 \times 0.5 = 5. The side adjacent = 10×cos⁡30°=10×0.866=8.6610 \times \cos 30° = 10 \times 0.866 = 8.66.
Common angle values
At 45°: sin⁡45°=cos⁡45°=22≈0.707\sin 45° = \cos 45° = \frac{\sqrt{2}}{2} \approx 0.707. At 60°: sin⁡60°=32\sin 60° = \frac{\sqrt{3}}{2}, cos⁡60°=12\cos 60° = \frac{1}{2}, tan⁡60°=3\tan 60° = \sqrt{3}.

Common Confusion Points

The most common mistake is mixing up which side is 'opposite' and which is 'adjacent.' Always identify them relative to the angle you're working with — the opposite side is across from the angle, the adjacent side is next to it (but not the hypotenuse).
On the unit circle, sine gives the y-coordinate and cosine gives the x-coordinate of the point. Many students confuse which is which — remember: cosine comes first alphabetically, just like x comes before y.

Frequently Asked Questions

What is SOHCAHTOA?
SOHCAHTOA is a mnemonic: SOH = Sine is Opposite over Hypotenuse, CAH = Cosine is Adjacent over Hypotenuse, TOA = Tangent is Opposite over Adjacent. It only applies to right triangles.
How are sine and cosine related?
Sine and cosine are complementary: sin(θ) = cos(90° − θ). They also satisfy the Pythagorean identity: sin²θ + cos²θ = 1. On a graph, cosine is just sine shifted left by 90° (π/2 radians).
When is tangent undefined?
Tangent is undefined when cosine equals zero, which happens at 90° and 270° (and every 180° from there). At these angles, the tangent line is vertical — its slope is undefined.

Related Pages