Skill: Determine cosine ratios in right triangles

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

$$\cos\theta=\frac{\text{adjacent leg}}{\text{hypotenuse}}$$

• The hypotenuse is the side opposite the right angle and is always the longest side.
• The adjacent leg touches the angle $\theta$ but is not the hypotenuse.
• Do not use the leg opposite $\theta$.

Example: A right triangle has an acute angle of $35^\circ$. The leg adjacent to the angle is $8$ cm. Find the hypotenuse, $x$.

1. Identify the sides:

   • Adjacent leg: $8$ cm
   • Hypotenuse: $x$

2. Write the cosine equation:

$$\cos 35^\circ=\frac{8}{x}$$

3. Solve for $x$ by multiplying both sides by $x$:

$$x\cos35^\circ=8$$

4. Divide by $\cos35^\circ$:

$$x=\frac{8}{\cos35^\circ}$$

5. Use a calculator in degree mode:

$$x\approx\frac{8}{0.8192}\approx9.77$$

Therefore, the hypotenuse is approximately

$$\boxed{9.77\text{ cm}}$$

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