Question

Using the 30°-60°-90°triangle shown on page 703, verify each value.

a. $\sin 30{}^{\circ }=\frac{1}{2}$

b. $\cos 30{}^{\circ }=\frac{\sqrt{3}}{2}$

c. $\sin 60{}^{\circ }=\frac{\sqrt{3}}{2}$

Short answer

1. The verified value is$\sin 30{}^{\circ }=\frac{1}{2}$.
2. The verified value is$\cos 30{}^{\circ }=\frac{\sqrt{3}}{2}$.
3. The verified value is$\sin 60{}^{\circ }=\frac{\sqrt{3}}{2}$.

Step-by-step solution

a.Step 1. Given information.

Given to use the 30°-60°-90° triangle below to verify the value $\sin 30{}^{\circ }=\frac{1}{2}$

Step 2. Explanation.

The sine function of an angle is defined as the ratio between the opposite side and hypotenuse of the angle of the right triangle:

$\sin \theta =\frac{\text{opp}}{\text{hyp}}$

The side opposite to 30° is x and the hypotenuse is 2x.

Plugging the values from the triangle:

$\begin{array}{l}\sin 30°=\frac{x}{2x}\\ \sin 30°=\frac{1}{2}\end{array}$

The value matches with the given value i.e. $\sin 30{}^{\circ }=\frac{1}{2}$

Step 3. Conclusion.

Hence the verified value is $\sin 30{}^{\circ }=\frac{1}{2}$.

b.Step 1. Given information.

Given to use the 30°-60°-90° triangle below to verify the value $\cos 30{}^{\circ }=\frac{\sqrt{3}}{2}$

Step 2. Explanation.

The cosine function of an angle is defined as the ratio between the adjacent side and hypotenuse of the angle of the right triangle:

$\cos \theta =\frac{\text{adj}}{\text{hyp}}$

The side adjacent to 30° is x√3 and the hypotenuse is 2x.

Plugging the values from the triangle:

$\begin{array}{l}\cos 30°=\frac{x\sqrt{3}}{2x}\\ \cos 30°=\frac{\sqrt{3}}{2}\end{array}$

The value matches with the given value i.e. $\cos 30{}^{\circ }=\frac{\sqrt{3}}{2}$

Step 3. Conclusion.

Hence the verified value is $\cos 30{}^{\circ }=\frac{\sqrt{3}}{2}$.

c.Step 1. Given information.

Given to use the 30°-60°-90° triangle below to verify the value $\sin 60{}^{\circ }=\frac{\sqrt{3}}{2}$

Step 2. Explanation.

The sine function of an angle is defined as the ratio between the opposite side and hypotenuse of the angle of the right triangle:

$\sin \theta =\frac{\text{opp}}{\text{hyp}}$

The side opposite to 60° is x√3 and the hypotenuse is 2x.

Plugging the values from the triangle:

$\begin{array}{l}\sin 60°=\frac{x\sqrt{3}}{2x}\\ \sin 60°=\frac{\sqrt{3}}{2}\end{array}$

The value matches with the given value i.e. $\sin 60{}^{\circ }=\frac{\sqrt{3}}{2}$

Step 3. Conclusion.

Hence the verified value is $\sin 60{}^{\circ }=\frac{\sqrt{3}}{2}$.