Sine Ratio (SOH)

Master the sine ratio and learn how to find missing sides and angles using opposite and hypotenuse.

Advanced25 minLesson
Printable worksheet

Definition

The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.
sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
The mnemonic SOH helps you remember:
  • S = Sine
  • O = Opposite
  • H = Hypotenuse
For example, in a right triangle where the angle is 30°30°:
sin⁡(30°)=12=0.5\sin(30°) = \frac{1}{2} = 0.5
This means the opposite side is exactly half the length of the hypotenuse.

Try it now

In a right triangle, the sine of an angle equals:

Worked Examples

A ladder leans against a wall at a 60°60° angle with the ground. If the ladder is 1010 meters long, how high up the wall does it reach?
1

Identify the known values

Angle = 60°60°, Hypotenuse (ladder) = 1010 m, Unknown = Opposite (height) → Set up the problem
2

Write the sine formula

sin⁡(60°)=opposite10\sin(60°) = \frac{\text{opposite}}{10} → Formula ready
3

Find sin⁡(60°)\sin(60°)

sin⁡(60°)=32≈0.866\sin(60°) = \frac{\sqrt{3}}{2} \approx 0.866
4

Solve for the opposite side

opposite=10×0.866=8.66\text{opposite} = 10 \times 0.866 = 8.66 m → 8.668.66 meters

Common Mistakes

Confusing opposite and adjacent sides

Why it's wrong: The opposite and adjacent sides depend on which angle you're working with. The opposite side is always across from your angle.

Correct: Always identify your reference angle first, then label: opposite is across from it, adjacent is next to it (not the hypotenuse).

Using sine when you should use a different ratio

Why it's wrong: Sine only involves the opposite and hypotenuse. If you know the adjacent side, you need cosine or tangent.

Correct: Check which sides you know: Opposite + Hypotenuse → Sine, Adjacent + Hypotenuse → Cosine, Opposite + Adjacent → Tangent.

Forgetting to check calculator mode (degrees vs radians)

Why it's wrong: Most school problems use degrees, but calculators might be set to radians. sin⁡(30)\sin(30) in radians gives a completely different answer!

Correct: Always verify your calculator is in degree mode (DEG) when working with degree measures.

Thinking sin⁡(2×30°)=2×sin⁡(30°)\sin(2 \times 30°) = 2 \times \sin(30°)

Why it's wrong: Sine is not a linear function. sin⁡(60°)≠2×sin⁡(30°)\sin(60°) \neq 2 \times \sin(30°).

Correct: sin⁡(60°)=32≈0.866\sin(60°) = \frac{\sqrt{3}}{2} \approx 0.866, while 2×sin⁡(30°)=2×0.5=12 \times \sin(30°) = 2 \times 0.5 = 1. Always calculate directly.

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

Explore how the sine ratio (opposite/hypotenuse) changes with the angle.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
In a right triangle, the sine of an angle equals:

Why It Matters

The sine ratio is one of the most powerful tools in mathematics and has countless real-world applications:
  • Architecture: Calculating roof pitch and building heights
  • Navigation: Determining distances and bearings for ships and aircraft
  • Physics: Analyzing projectile motion and wave behavior
  • Engineering: Designing ramps, bridges, and mechanical systems
  • Astronomy: Measuring distances to stars and planets
Any time you need to find an unknown length or angle in a right triangle, sine (along with cosine and tangent) is your key to the solution.

Real World Applications

Finding Building Heights

Surveyors use the sine ratio to calculate the height of tall structures without climbing them.

Example:

Standing 50 meters from a building, you measure an angle of elevation of 65°65° to the top. Using sine in the calculations helps determine the building's height.

1Try It Yourself

A tree casts a shadow. You stand at the end of the shadow and measure the angle to the treetop as 40°40°. The straight-line distance from you to the treetop is 2525 meters.

How tall is the tree?

Step 1: Write the mathematical expression

Use sine: height = 25×sin⁡(40°)25 \times \sin(40°)

Aviation and Flight Paths

Pilots use trigonometric ratios to calculate climb rates and approach angles.

Example:

When an airplane climbs at an angle of 15°15° and travels 2 kilometers along its flight path, sine tells us how much altitude it gains.

2Try It Yourself

An airplane takes off and climbs at an angle of 12°12°. After traveling 33 km along its flight path (hypotenuse), how high is the plane?

What is the plane's altitude?

Step 1: Write the mathematical expression

Altitude = 3×sin⁡(12°)3 \times \sin(12°)

Key Takeaways

  • 1Sine equals opposite divided by hypotenuse: sin⁡(θ)=opphyp\sin(\theta) = \frac{\text{opp}}{\text{hyp}}
  • 2Remember SOH: Sine = Opposite / Hypotenuse
  • 3To find the opposite: multiply hypotenuse by sin⁡(θ)\sin(\theta)
  • 4To find the hypotenuse: divide opposite by sin⁡(θ)\sin(\theta)
  • 5To find the angle: use inverse sine θ=sin⁡−1(opphyp)\theta = \sin^{-1}\left(\frac{\text{opp}}{\text{hyp}}\right)
  • 6Special values: sin⁡(30°)=0.5\sin(30°) = 0.5, sin⁡(45°)=22\sin(45°) = \frac{\sqrt{2}}{2}, sin⁡(60°)=32\sin(60°) = \frac{\sqrt{3}}{2}, sin⁡(90°)=1\sin(90°) = 1

Frequently Asked Questions

The word 'sine' comes from the Latin 'sinus' meaning 'bay' or 'fold.' It was a translation from Arabic 'jayb,' which itself was a transliteration of the Sanskrit 'jya' meaning 'bowstring' - referring to the half-chord of a circle.
The word 'sine' comes from the Latin 'sinus' meaning 'bay' or 'fold.' It was a translation from Arabic 'jayb,' which itself was a transliteration of the Sanskrit 'jya' meaning 'bowstring' - referring to the half-chord of a circle.
The sine of any angle is always between −1-1 and 11. The maximum value is 11, which occurs at 90°90° (or π2\frac{\pi}{2} radians). This makes sense because the opposite side can never be longer than the hypotenuse.
Use sine when you're working with the opposite side and the hypotenuse. If you have (or need) the adjacent side, use cosine (with hypotenuse) or tangent (with opposite).
sin⁡−1\sin^{-1} is the inverse sine function, also called arcsine (arcsin). It answers: 'What angle has this sine value?' For example, sin⁡−1(0.5)=30°\sin^{-1}(0.5) = 30° because sin⁡(30°)=0.5\sin(30°) = 0.5.

Glossary

Sine
The ratio of the opposite side to the hypotenuse in a right triangle: sin⁡(θ)=opphyp\sin(\theta) = \frac{\text{opp}}{\text{hyp}}
Opposite side
The side of a right triangle that is across from (opposite to) the reference angle
Hypotenuse
The longest side of a right triangle, always opposite the right angle
Inverse sine
The function sin⁡−1\sin^{-1} (arcsin) that finds an angle when you know its sine value
SOH CAH TOA
Memory aid for trig ratios: Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent

Formula Card

Sine definition

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

The ratio of the opposite side to the hypotenuse

Find opposite

opposite=hypotenuse×sin⁡(θ)\text{opposite} = \text{hypotenuse} \times \sin(\theta)

Multiply hypotenuse by sine to find the opposite side

Find hypotenuse

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

Divide opposite by sine to find the hypotenuse

Find angle

θ=sin⁡−1(oppositehypotenuse)\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)

Use inverse sine (arcsin) to find the angle

Special value: sin(30)

sin⁡(30°)=12=0.5\sin(30°) = \frac{1}{2} = 0.5

Sine of 30 degrees equals one-half

Special value: sin(45)

sin⁡(45°)=22≈0.707\sin(45°) = \frac{\sqrt{2}}{2} \approx 0.707

Sine of 45 degrees equals square root of 2 over 2

Special value: sin(60)

sin⁡(60°)=32≈0.866\sin(60°) = \frac{\sqrt{3}}{2} \approx 0.866

Sine of 60 degrees equals square root of 3 over 2

Today's challengeOne problem, three tries

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