Introduction to Trigonometric Ratios

Learn the three fundamental trigonometric ratios (sine, cosine, tangent) and the SOH-CAH-TOA mnemonic.

Advanced25 minLesson
Printable worksheet

Definition

Trigonometric ratios are relationships between the sides of a right triangle and its angles. In any right triangle, we can define three primary ratios based on a chosen acute angle:
SOH-CAH-TOA is the mnemonic to remember these ratios:
sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}
Key terms:
  • Opposite: The side across from the angle θ\theta
  • Adjacent: The side next to the angle θ\theta (not the hypotenuse)
  • Hypotenuse: The longest side, always opposite the right angle

Try it now

In SOH-CAH-TOA, what does 'SOH' stand for?

Worked Examples

In a right triangle with angle θ\theta at vertex A, identify the opposite, adjacent, and hypotenuse sides.
1

Find the hypotenuse

The hypotenuse is always the longest side, opposite the right angle (90 degrees) → Side across from the right angle
2

Find the opposite side

The opposite side is directly across from angle θ\theta - it does not touch angle θ\theta → Side facing angle θ\theta
3

Find the adjacent side

The adjacent side touches angle θ\theta but is not the hypotenuse → Side next to angle θ\theta

Common Mistakes

Confusing opposite and adjacent sides

Why it's wrong: The labels 'opposite' and 'adjacent' depend on which angle you're looking at. When the reference angle changes, so do the labels.

Correct: Always identify the angle first, then label: opposite is across from that angle, adjacent touches that angle (but isn't the hypotenuse).

Using the wrong ratio formula

Why it's wrong: Students often mix up which sides go in the numerator and denominator.

Correct: Use SOH-CAH-TOA: Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent.

Forgetting that these ratios only work for right triangles

Why it's wrong: The definitions of opposite, adjacent, and hypotenuse require a 90-degree angle.

Correct: Verify the triangle has a right angle before applying SOH-CAH-TOA. For other triangles, use the Law of Sines or Cosines.

Thinking sin⁡(θ)\sin(\theta) can be greater than 1

Why it's wrong: Since the hypotenuse is always the longest side, OppositeHypotenuse\frac{\text{Opposite}}{\text{Hypotenuse}} can never exceed 1.

Correct: In a right triangle, the sine and cosine of an acute angle are always between 0 and 1. If you get a value greater than 1, you probably divided by a leg instead of the hypotenuse.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Six Trigonometric Functions Explained | Sine, Cosine, Tangent, Secant, Cosecant, Cotangent

by Understand The Math

Watch on YouTube

Interactive Visual

Right Triangle Trigonometry

θ =30°
5°45°85°
sin(θ)
0.5
Opp / Hyp
cos(θ)
√3/2
Adj / Hyp
tan(θ)
√3/3
Opp / Adj

Remember: SOH-CAH-TOA

SOH

Sin = Opp / Hyp

CAH

Cos = Adj / Hyp

TOA

Tan = Opp / Adj

Click on sin, cos, or tan to highlight the relevant sides of the triangle.

Triangle Explorer

a = 3b = 4c = 5

a² + b² = c²

3² + 4² = 5²

9 + 16 = 25

Change the leg lengths to see the Pythagorean theorem in action.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
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In SOH-CAH-TOA, what does 'SOH' stand for?

Why It Matters

Trigonometric ratios are foundational tools used across many fields:
  • Architecture & Construction: Calculating roof angles, ramp slopes, and structural supports
  • Navigation: Determining distances and directions for ships, planes, and GPS systems
  • Physics: Analyzing forces, projectile motion, and wave behavior
  • Astronomy: Measuring distances to stars and the size of celestial objects
  • Video Games & Animation: Creating realistic 3D graphics and character movements
Understanding these ratios unlocks the ability to solve problems involving angles and distances without measuring directly!

Real World Applications

Measuring Building Heights

Surveyors use trigonometry to measure the height of buildings without climbing them.

Example:

Standing 50 meters from a building, a surveyor measures the angle to the top as 60 degrees. Using tan⁡(60°)≈1.73\tan(60°) \approx 1.73, the building height is approximately 50×1.73=86.550 \times 1.73 = 86.5 meters.

1Try It Yourself

You stand 30 meters from a tree. The angle from your eye level to the treetop is 45 degrees.

How tall is the tree above your eye level?

Step 1: Write the mathematical expression

Use tangent: height = distance ×\times tan(angle)

Calculating Roof Pitch

Builders use trigonometry to determine the angle and slope of roofs for proper water drainage.

Example:

A roof rises 4 meters over a horizontal span of 6 meters. The pitch angle is arctan⁡(4/6)≈33.7°\arctan(4/6) \approx 33.7°.

2Try It Yourself

A roof rises 5 meters vertically over a horizontal distance of 12 meters.

What is the tangent of the roof angle?

Step 1: Write the mathematical expression

Calculate: rise / run

Navigation and Aviation

Pilots use trigonometry to calculate descent angles and distance to runways.

Example:

A plane at 900 meters altitude needs to descend at a 3-degree angle. Using tan⁡(3°)≈0.0524\tan(3°) \approx 0.0524, the plane should begin descent about 9000.0524≈17,200\frac{900}{0.0524} \approx 17{,}200 meters (about 17 km) from the runway.

Key Takeaways

  • 1Trigonometric ratios relate angles to side lengths in right triangles
  • 2SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
  • 3The hypotenuse is always the longest side, opposite the right angle
  • 4Opposite and adjacent sides depend on which angle you're measuring from
  • 5For an acute angle in a right triangle, sine and cosine are always between 0 and 1; tangent can be any positive number

Frequently Asked Questions

SOH-CAH-TOA is a mnemonic (memory aid) where each three-letter group represents a ratio: SOH = Sine is Opposite over Hypotenuse, CAH = Cosine is Adjacent over Hypotenuse, TOA = Tangent is Opposite over Adjacent.
SOH-CAH-TOA is a mnemonic (memory aid) where each three-letter group represents a ratio: SOH = Sine is Opposite over Hypotenuse, CAH = Cosine is Adjacent over Hypotenuse, TOA = Tangent is Opposite over Adjacent.
The basic SOH-CAH-TOA definitions only work for right triangles. For other triangles, you need the Law of Sines or Law of Cosines.
Use inverse trigonometric functions: θ=arcsin⁡(x)\theta = \arcsin(x), θ=arccos⁡(x)\theta = \arccos(x), or θ=arctan⁡(x)\theta = \arctan(x). These are also written as sin⁡−1\sin^{-1}, cos⁡−1\cos^{-1}, and tan⁡−1\tan^{-1}.

Glossary

Trigonometric ratio
A ratio comparing two sides of a right triangle, determined by an angle
Sine (sin)
The ratio of the opposite side to the hypotenuse: sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
Cosine (cos)
The ratio of the adjacent side to the hypotenuse: cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
Tangent (tan)
The ratio of the opposite side to the adjacent side: tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
Hypotenuse
The longest side of a right triangle, located opposite the right angle
Opposite side
The side of a right triangle that is across from (does not touch) the reference angle
Adjacent side
The side of a right triangle that touches the reference angle and is not the hypotenuse

Formula Card

Sine

sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

SOH - ratio of the side opposite angle theta to the hypotenuse

Cosine

cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

CAH - ratio of the side adjacent to angle theta to the hypotenuse

Tangent

tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

TOA - ratio of the opposite side to the adjacent side

Today's challengeOne problem, three tries

More in This Topic

Sine Ratio (SOH)

Advanced25 min

Cosine Ratio (CAH)

Advanced25 min

Tangent Ratio (TOA)

Advanced25 min