Sine Ratio (SOH)
Ratio of the side opposite an angle to the hypotenuse.
Formula
Variables
Example
sin 30° = 0.5
How it works
In a right triangle, the size of an acute angle fixes the proportions of its sides, so the sine of that angle is defined as the length of the side opposite the angle divided by the hypotenuse. This gives a consistent way to measure angles using side lengths, and it stays the same for any similar right triangle because all corresponding sides scale by the same factor. That is why sine is a ratio rather than a raw length.
When to use it
Use this ratio in right-triangle problems when you know an angle and the hypotenuse or when you need to find the side opposite an angle. It is common in geometry, navigation, surveying, physics, and any situation involving heights, slopes, or components of motion.
Worked example, step by step
A right triangle has a hypotenuse of 12 cm and an angle of 25°. Step 1: Use the sine ratio: sin θ = opposite / hypotenuse Step 2: Substitute the known values: sin 25° = opposite / 12 Step 3: Use a calculator: sin 25° ≈ 0.4226 Step 4: Solve for the opposite side: opposite = 12 × 0.4226 Step 5: Calculate: opposite ≈ 5.07 cm Final answer: the side opposite the 25° angle is about 5.1 cm.
Common mistakes to avoid
- Using the adjacent side instead of the opposite side
- Putting the hypotenuse in the wrong position in the fraction
- Forgetting that sine applies to right triangles in basic trigonometry
Did You Know?
The word "sine" is a mistranslation: Arabic "jayb" (bay) was misread, and the Latin "sinus" means a fold or bay.
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More in Trigonometry
View allCosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Cosecant
IntermediateReciprocal of sine.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.
Tangent Identity
BasicTangent equals sine divided by cosine.