Mathwords logoMathwords

Trigonometric Ratios — Definition, Formula & Examples

Trigonometric ratios are the ratios of two sides of a right triangle relative to one of its acute angles. The three primary ratios are sine, cosine, and tangent, and each one pairs a specific combination of the opposite side, adjacent side, or hypotenuse.

Given a right triangle with an acute angle θ\theta, the trigonometric ratios are defined as sin⁡θ=oppositehypotenuse\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}, cos⁡θ=adjacenthypotenuse\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}, and tan⁡θ=oppositeadjacent\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}. Three reciprocal ratios—cosecant, secant, and cotangent—are defined as csc⁡θ=1sin⁡θ\csc\theta = \dfrac{1}{\sin\theta}, sec⁡θ=1cos⁡θ\sec\theta = \dfrac{1}{\cos\theta}, and cot⁡θ=1tan⁡θ\cot\theta = \dfrac{1}{\tan\theta}, respectively.

Key Formula

sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}
Where:
  • θ\theta = An acute angle in the right triangle
  • opp\text{opp} = Length of the side opposite angle θ
  • adj\text{adj} = Length of the side adjacent to angle θ (not the hypotenuse)
  • hyp\text{hyp} = Length of the hypotenuse (the side opposite the right angle)

How It Works

To use a trigonometric ratio, first identify the acute angle of interest in a right triangle. Then label the three sides relative to that angle: the side directly across from it is the opposite, the side touching it (that is not the hypotenuse) is the adjacent, and the longest side across from the right angle is the hypotenuse. Choose the ratio that connects the two sides you know or need. Set up an equation and solve for the unknown side length or angle. When solving for an angle, you apply inverse trigonometric functions to the ratio's value.

Worked Example

Problem: A right triangle has a hypotenuse of 10 and an acute angle of 30°. Find the lengths of the opposite and adjacent sides.
Find the opposite side: The opposite side relates to the hypotenuse through sine.
sin⁡30°=opp10  ⟹  opp=10⋅sin⁡30°=10⋅0.5=5\sin 30° = \frac{\text{opp}}{10} \implies \text{opp} = 10 \cdot \sin 30° = 10 \cdot 0.5 = 5
Find the adjacent side: The adjacent side relates to the hypotenuse through cosine.
cos⁡30°=adj10  ⟹  adj=10⋅cos⁡30°=10⋅32=53≈8.66\cos 30° = \frac{\text{adj}}{10} \implies \text{adj} = 10 \cdot \cos 30° = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66
Verify with the Pythagorean theorem: Check that the two sides and hypotenuse satisfy the theorem.
52+(53)2=25+75=100=102  ✓5^2 + (5\sqrt{3})^2 = 25 + 75 = 100 = 10^2 \;\checkmark
Answer: The opposite side is 5 and the adjacent side is 53≈8.665\sqrt{3} \approx 8.66.

Another Example

Problem: A ladder leans against a wall. Its foot is 6 m from the base of the wall, and the ladder makes a 55° angle with the ground. How high up the wall does the ladder reach?
Identify the ratio: The 6 m distance is adjacent to the 55° angle, and the wall height is opposite. Use tangent.
tan⁡55°=height6\tan 55° = \frac{\text{height}}{6}
Solve for height: Multiply both sides by 6 and evaluate.
height=6⋅tan⁡55°≈6⋅1.4281≈8.57 m\text{height} = 6 \cdot \tan 55° \approx 6 \cdot 1.4281 \approx 8.57 \text{ m}
Answer: The ladder reaches approximately 8.57 m up the wall.

Visualization

Why It Matters

Trigonometric ratios appear throughout high-school Geometry (standard G-SRT.C.6) and are prerequisite knowledge for Precalculus, Calculus, and Physics. Surveyors use them to calculate distances that are impossible to measure directly, and engineers rely on them to resolve forces into components. Mastering these ratios builds the foundation for understanding periodic functions, wave behavior, and circular motion.

Common Mistakes

Mistake: Mixing up opposite and adjacent sides when the reference angle changes.
Correction: Always label sides relative to the specific angle you are working with. The opposite and adjacent sides swap when you switch to the other acute angle in the same triangle.
Mistake: Using trigonometric ratios with a calculator set to the wrong angle mode (radians instead of degrees, or vice versa).
Correction: Check your calculator's mode before computing. For a 30° angle, your calculator must be in degree mode; in radian mode you would enter π/6 instead.

Related Terms