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.

The mnemonic SOH helps you remember:

S = Sine
O = Opposite
H = Hypotenuse

For example, in a right triangle where the angle is :

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 angle with the ground. If the ladder is meters long, how high up the wall does it reach?

Identify the known values

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

Write the sine formula

→ Formula ready

Find

→

Solve for the opposite side

m → meters

Answer: The ladder reaches approximately meters up the wall.

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. in radians gives a completely different answer!

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

Thinking

Why it's wrong: Sine is not a linear function. .

Correct: , while . 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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History

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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 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 . The straight-line distance from you to the treetop is meters.

How tall is the tree?

Step 1: Write the mathematical expression

Use sine: height =

Aviation and Flight Paths

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

Example:

When an airplane climbs at an angle of 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 . After traveling km along its flight path (hypotenuse), how high is the plane?

What is the plane's altitude?

Step 1: Write the mathematical expression

Altitude =

Key Takeaways

1Sine equals opposite divided by hypotenuse:
2Remember SOH: Sine = Opposite / Hypotenuse
3To find the opposite: multiply hypotenuse by
4To find the hypotenuse: divide opposite by
5To find the angle: use inverse sine
6Special values: , , ,

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.

Glossary

Sine: The ratio of the opposite side to the hypotenuse in a right triangle:
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 (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

The ratio of the opposite side to the hypotenuse

Find opposite

Multiply hypotenuse by sine to find the opposite side

Find hypotenuse

Divide opposite by sine to find the hypotenuse

Find angle

Use inverse sine (arcsin) to find the angle

Special value: sin(30)

Sine of 30 degrees equals one-half

Special value: sin(45)

Sine of 45 degrees equals square root of 2 over 2

Special value: sin(60)

Sine of 60 degrees equals square root of 3 over 2