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Determine sine ratios in right triangles

For an acute angle in a right triangle, the sine ratio is the length of the side opposite the angle divided by the hypotenuse: sin⁡θ=oppositehypotenuse\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}. The ratio is invariant across similar right triangles and changes when the reference angle changes, supporting calculation of unknown side lengths and angle measures; this scope is limited to acute angles in right triangles and excludes unit-circle definitions, obtuse angles, graphs, and trigonometric identities.

Detailed Explanation: Determine sine ratios in right triangles

For an acute angle in a right triangle, use the sine ratio:

sin⁡θ=length of the opposite sidelength of the hypotenuse\sin\theta=\frac{\text{length of the opposite side}}{\text{length of the hypotenuse}}
  • Opposite: the side directly across from the chosen angle
  • Hypotenuse: the side across from the right angle; it is always the longest side

Example: In right triangle (ABC)(ABC), ∠C=90∘\angle C=90^\circ. For ∠A\angle A, the side opposite is (BC=6)(BC=6) cm, and the hypotenuse is (AB=10)(AB=10) cm. Find sin⁡A\sin A.

  1. Identify the reference angle: ∠A\angle A.
  2. Identify the opposite side: (BC=6)(BC=6) cm.
  3. Identify the hypotenuse: (AB=10)(AB=10) cm.
  4. Substitute into the sine formula:
sin⁡A=oppositehypotenuse=610\sin A=\frac{\text{opposite}}{\text{hypotenuse}} =\frac{6}{10}
  1. Simplify:
sin⁡A=35=0.6\boxed{\sin A=\frac{3}{5}=0.6}

Always choose the sides based on the angle you are using. If you used ∠B\angle B instead, the opposite side would be different, so the sine ratio would change.

Learn by doing: Determine sine ratios in right triangles

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