The Cosine Ratio

Worked example

1. 1

   $\cos 32^{\circ }=\frac{x}{9}$ — x touches the 32° angle, so it is the adjacent. The 9 cm side is the longest, the hypotenuse. Adjacent and hypotenuse mean cos.

2. 2

   $x=9\times \cos 32^{\circ }$ — x is divided by 9, so multiply both sides by 9 to leave x alone.

3. 3

   $x=7.632\dots$ — Type 9 × cos 32 into the calculator, set to degrees.

4. 4

   $x=7.63\text{ cm}$ — Round to 3 significant figures and add the unit. It is shorter than the 9 cm hypotenuse, as it should be.

Worked example

1. 1

   $\cos 41^{\circ }=\frac{8.5}{x}$ — The 8.5 cm side touches the 41° angle, so it is the adjacent. x is the hypotenuse. Adjacent and hypotenuse mean cos.

2. 2

   $x\times \cos 41^{\circ }=8.5$ — x is on the bottom of the fraction. Multiply both sides by x to bring it up.

3. 3

   $x=\frac{8.5}{\cos 41^{\circ }}$ — x is now multiplied by cos 41°. Divide both sides by cos 41° to leave x alone.

4. 4

   $x=11.26\dots$ — Type 8.5 ÷ cos 41 into the calculator.

5. 5

   $x=11.3\text{ cm}$ — Round to 3 significant figures. It is longer than 8.5 cm, as a hypotenuse must be.

Worked example

1. 1

   $\cos x=\frac{6.4}{10}$ — The 6.4 cm side touches the angle, so it is the adjacent. The 10 cm side is the hypotenuse. Adjacent and hypotenuse mean cos.

2. 2

   $\cos x=0.64$ — Divide 6.4 by 10. The answer is below 1, as a cosine must be.

3. 3

   $x={\cos }^{-1}(0.64)$ — We know the cosine and want the angle. Inverse cosine, ${\cos }^{-1}$, undoes cos.

4. 4

   $x=50.21\dots$ — Type the ${\cos }^{-1}$ of 0.64 into the calculator.

5. 5

   $x=50.2^{\circ }$ — Round to 1 decimal place, the usual accuracy for angles, and add the degree sign.