Question

Hello can you help me with this trigonometry problem and this a homework assignment

Answer 1

To determine the missing measurements of the given right triangle you have to apply the trigonometric ratios:

`\begin{gathered} \sin \theta=(opposite)/(hypothenuse) \\ \cos \theta=(adjacent)/(hypothenuse) \\ \tan \theta=(opposite)/(adjacent) \end{gathered}`

First determine the measure of angle B

Remember that the sum of the inner angles of any triangle is 180º, we knw two out of the three angles, so we can calculate the missing one as follows:

`\begin{gathered} 30+90+B=180 \\ 120+B=180 \\ B=180-120 \\ B=60 \end{gathered}`

Angle B=60º

Next is the hypothenuse "h"

We know angle θ=30º and its opposite side 9, using the sine we can calculate the measurement of the hypothenuse as:

`\begin{gathered} \sin \theta=(opposite)/(hypothenuse) \\ \sin 30=(9)/(h) \\ h\sin 30=9 \\ h=(9)/(\sin 30) \\ h=18 \end{gathered}`

The hypothenuse of this triangle is h=18

Now that we know the value of the hypothenuse, we can apply the Pythagoras theorem to determine the value of s.

Remember tha this theorem states that the square of the hypothenuse is equal to the sum of the squares of the base and height, symbolically:

`a^2+b^2=c^2`

For this exercise c=h=18

a=9

b=s=?

Lets replace and solve:

`\begin{gathered} 9^2+s^2=18^2 \\ 81+s^2=324 \\ s^2=324-81 \\ s^2=243 \\ s=\sqrt[]{243} \\ s=9\sqrt[]{3}\cong15.59 \end{gathered}`

You can also calculate the value of s using the trigonometric ratios, for θ=30º, s is its adjacent side and 9is the opposite side. The ratio that relates the adjacent side of the angle with its opposite side is tha tangent:

`\begin{gathered} \tan \theta=(opposite)/(adjacent) \\ \tan 30=(9)/(s) \\ s\tan 30=9 \\ s=(9)/(\tan 30) \\ s=9\sqrt[]{3}\cong15.59 \end{gathered}`

Using either method is valid.

So, the missing measurements are:

∠B=60º

h=18

s=9√3