Question

For the right triangles below, find the exact values of the side lengths b and h. If necessary, write your responses in simplified radical form. b = 300 Х $ ? b 45° 8 h = 60° 459 5 7

Answer 1

`\begin{gathered} b\text{ = }\frac{7\sqrt[]{3}}{2} \\ \\ h\text{ =4}\sqrt[]{2} \end{gathered}`

Here, we want to find the exact values of the side lengths b and h

For b, we have to consider its triangle

The side b faces the angle 30 and thus it is opposite to it

The side 7 is adjacent to the angle 30

We have to use the trigonometric identity that relates adjacent to opposite

The trigonometric identity that does this is the cosine

The cosine of an angle is the ratio of opposite to its adjacent

Mathematically;

`\begin{gathered} \cos \text{ 30 = }(b)/(7) \\ \\ b\text{ = 7 cos 30} \\ \\ b\text{ = 7 }*\text{ }\frac{\sqrt[]{3}}{2} \\ \\ b\text{ = }\frac{7\sqrt[]{3}}{2} \end{gathered}`

For h, we need to apply the appropriate trigonometric identity

As we can see, the angle 45 faces the side h, this means it is opposite

8 represents the hypotenuse

The relationship between opposite and hypotenuse is shown through sine

The sine of an angle is the ratio of the opposite to the hypotenuse

Thus, mathematically;

`\begin{gathered} \sin \text{ 45 = }(h)/(8) \\ \\ h\text{ = 8 sin 45} \\ \\ h\text{ = 8 }*\text{ }\frac{1}{\sqrt[]{2}\text{ }}\text{ = 4}\sqrt[]{2} \end{gathered}`