Key takeaway: The sine of an acute angle in a right triangle is the ratio of the length of the side opposite that angle to the length of the hypotenuse.
Grade 10 – GeometryG-SRT.C.6Trigonometry
Definition of Sine
The sine of an angle divides one side of a right triangle by another: the side across from the angle over the longest side. The answer is a pure number, with no units attached.
Right triangles sharing an angle are scaled copies of each other, so the quotient is set by the angle alone. One calculator key therefore serves every triangle. The longest side is the divisor, which keeps results for sharp angles strictly between 0 and 1; they climb steadily as the angle widens toward a right angle.
Formula and notation
Written \(\sin\theta\), and read “sine of theta”:
\[ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
To find an angle from the ratio, use the inverse key: \(\theta = \sin^{-1}(\text{ratio})\).
Key facts
- It is the S in SOH of the SOH-CAH-TOA memory aid.
- Sharp-angle results are always proper fractions: positive and below 1.
- It equals the cosine of the complementary angle: sin 25 degrees equals cos 65 degrees.
- On a calculator, the sin button needs degree mode for angles in degrees.
Sine example
A loading ramp is 20 ft long and lifts the cargo 3.5 ft above the ground. What angle does it make with the ground? Then, for another ramp inclined at \(28^{\circ}\) and 25 ft long, find its rise. Use \(\sin 28^{\circ} \approx 0.4695\).
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First ramp: the rise faces the angle and the slanted surface is the longest side, so \(\sin\theta = \frac{3.5}{20} = 0.175\).
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Use the inverse key: \(\theta = \sin^{-1}(0.175) \approx 10.1^{\circ}\).
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Second ramp: \(\text{rise} = 25 \times \sin 28^{\circ} \approx 25 \times 0.4695 \approx 11.7\) ft.
The first ramp is inclined at about \(10.1^{\circ}\); the second rises about 11.7 ft.
Common mistake
Watch out
Hana computes the first ramp as \(\sin\theta = \frac{20}{3.5} \approx 5.71\) and then cannot find \(\sin^{-1}(5.71)\) on her calculator.
Why it is wrong
She inverted the ratio. Opposite goes on top and the hypotenuse on the bottom. A value above 1 can never occur for a sine, which is a signal that the fraction is upside down.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Trigonometry