Question

Using the 45°-45°-90° triangle shown on page 703, verify each value.

a. $\sin 45{}^{\circ }=\frac{\sqrt{2}}{2}$

b. $\cos 45{}^{\circ }=\frac{\sqrt{2}}{2}$

c. $\tan 45{}^{\circ }=1$

Short answer

1. The verified value is $\sin 45{}^{\circ }=\frac{\sqrt{2}}{2}$.
2. The verified value is $\cos 45{}^{\circ }=\frac{\sqrt{2}}{2}$.
3. The verified value is$\tan 45{}^{\circ }=1$.

Step-by-step solution

aStep 1. Given information.

Given to use the 45°-45°-90° triangle below to verify the value $\sin 45{}^{\circ }=\frac{\sqrt{2}}{2}$

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 45° is x and the hypotenuse is x√2.

Plugging the values from the triangle:

$\begin{array}{l}\sin 45°=\frac{x}{x\sqrt{2}}\\ \sin 45°=\frac{1}{\sqrt{2}}\\ \sin 45°=\frac{1\left(\sqrt{2}\right)}{\sqrt{2}\left(\sqrt{2}\right)}\\ \sin 45°=\frac{\sqrt{2}}{2}\end{array}$

The value matches with the given value i.e. $\sin 45{}^{\circ }=\frac{\sqrt{2}}{2}$

Step 3. Conclusion.

Hence the verified value is $\sin 45{}^{\circ }=\frac{\sqrt{2}}{2}$.

bStep 1. Given information.

Given to use the 45°-45°-90° triangle below to verify the value $\cos 45{}^{\circ }=\frac{\sqrt{2}}{2}$

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 45° is x and the hypotenuse is x√2.

Plugging the values from the triangle:

$\begin{array}{l}\cos 45°=\frac{x}{x\sqrt{2}}\\ \cos 45°=\frac{1}{\sqrt{2}}\\ \cos 45°=\frac{1\left(\sqrt{2}\right)}{\sqrt{2}\left(\sqrt{2}\right)}\\ \cos 45°=\frac{\sqrt{2}}{2}\end{array}$

The value matches with the given value i.e. $\cos 45{}^{\circ }=\frac{\sqrt{2}}{2}$

Step 3. Conclusion.

Hence the verified value is $\cos 45{}^{\circ }=\frac{\sqrt{2}}{2}$.

cStep 1. Given information.

Given to use the 45°-45°-90° triangle below to verify the value $\tan 45{}^{\circ }=1$

Step 2. Explanation.

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

$\tan \theta =\frac{\text{opp}}{\text{adj}}$

The side opposite to 45° is x and the adjacent is x.

Plugging the values from the triangle:

$\begin{array}{l}\tan 45°=\frac{x}{x}\\ \tan 45°=1\end{array}$

The value matches with the given value i.e. $\tan 45{}^{\circ }=1$

Step 3. Conclusion.

Hence the verified value is $\tan 45{}^{\circ }=1$.