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Determine cosine ratios in right triangles

For an acute angle in a right triangle, the cosine ratio is the length of the leg adjacent to the angle divided by the hypotenuse: cos⁡θ=adjacenthypotenuse\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}. This relationship is preserved in similar triangles and supports finding unknown side lengths or angle measures using appropriate numerical or calculator representations, while distinguishing the adjacent leg from the opposite leg; general-angle, unit-circle, and law-of-cosines treatments are not included.

Detailed Explanation: Determine cosine ratios in right triangles

For an acute angle θ\theta in a right triangle, use the cosine ratio:

cos⁡θ=adjacent leghypotenuse\cos\theta=\frac{\text{adjacent leg}}{\text{hypotenuse}}
  • The hypotenuse is the side opposite the right angle and is always the longest side.
  • The adjacent leg touches the angle θ\theta but is not the hypotenuse.
  • Do not use the leg opposite θ\theta.

Example: A right triangle has an acute angle of 35∘35^\circ. The leg adjacent to the angle is 88 cm. Find the hypotenuse, xx.

  1. Identify the sides:

    • Adjacent leg: 88 cm
    • Hypotenuse: xx

  2. Write the cosine equation:

cos⁡35∘=8x \cos 35^\circ=\frac{8}{x}

  1. Solve for xx by multiplying both sides by xx:

xcos⁡35∘=8 x\cos35^\circ=8

  1. Divide by cos⁡35∘\cos35^\circ:

x=8cos⁡35∘ x=\frac{8}{\cos35^\circ}

  1. Use a calculator in degree mode:

x≈80.8192≈9.77 x\approx\frac{8}{0.8192}\approx9.77

Therefore, the hypotenuse is approximately

9.77 cm\boxed{9.77\text{ cm}}

Always check that the hypotenuse is longer than the adjacent leg, which it is in this example.

Learn by doing: Determine cosine ratios in right triangles

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