The Sine Ratio

Worked example

1. 1

   $\sin 70^{\circ }=\frac{h}{6}$ — The height h is across from the 70° angle, so it is the opposite. The ladder is the longest side, the hypotenuse. Opposite and hypotenuse mean sin.

2. 2

   $h=6\times \sin 70^{\circ }$ — h is divided by 6, so multiply both sides by 6 to leave h on its own.

3. 3

   $h=5.638\dots$ — Type 6 × sin 70 into the calculator, set to degrees. Keep the whole number on the display for now.

4. 4

   $h=5.64\text{ m}$ — Round to 3 significant figures, the usual accuracy in exams, and put the unit on.

Worked example

1. 1

   $\sin 24^{\circ }=\frac{11}{x}$ — The 11 cm side is across from the 24° angle, so it is the opposite. x is the hypotenuse. Opposite and hypotenuse mean sin.

2. 2

   $x\times \sin 24^{\circ }=11$ — x is on the bottom of the fraction. Multiply both sides by x to bring it up.

3. 3

   $x=\frac{11}{\sin 24^{\circ }}$ — x is now multiplied by sin 24°. Divide both sides by sin 24° to leave x alone.

4. 4

   $x=27.04\dots$ — Type 11 ÷ sin 24 into the calculator.

5. 5

   $x=27.0\text{ cm}$ — Round to 3 significant figures. It is longer than 11 cm, as a hypotenuse must be.

Worked example

1. 1

   $\sin x=\frac{5}{13}$ — The 5 cm side is across from the angle, so it is the opposite. The 13 cm side is the longest, the hypotenuse. Opposite and hypotenuse mean sin.

2. 2

   $\sin x=0.3846\dots$ — Divide 5 by 13. The answer is below 1, as a sine must be.

3. 3

   $x={\sin }^{-1}(0.3846\dots )$ — We know the sine and want the angle. Inverse sine, ${\sin }^{-1}$, undoes sin.

4. 4

   $x=22.62\dots$ — Use the ${\sin }^{-1}$ key on the whole number from the calculator, not a rounded 0.385.

5. 5

   $x=22.6^{\circ }$ — Round to 1 decimal place, the usual accuracy for angles, and add the degree sign.