Question

Find the missing side of the triangle using the Pythagorean theorem and illustrate the six trigonometric ratios of θ.

Answer 1

Answer:

• The length of the missing side is 13.

,

• sin θ = 12/13, cos θ=5/13, tan θ = 12/5

,

• cosec θ = 13/12, sec θ = 13/5, cot θ = 5/12

Explanation:

Part A

First, we find the length of the missing side, MT using the Pythagorean Theorem.

$\begin{gathered} Hypotenuse^2=Altitude^2+Base^2 \\ |MT|^2=|MA|^2+|AT|^2 \end{gathered}$

Substitute the known values:

$\begin{gathered} |MT|^2=12^2+5^2 \\ \lvert MT\rvert=\sqrt{12^2+5^2}=\sqrt{169}=13 \end{gathered}$

The length of the missing side is 13.

Part B

Next, we find the six trigonometric ratios of θ.

• The side ,opposite angle ,θ = 12

,

• The side ,adjacent to ,angle θ = 5

,

• The ,hypotenuse ,= 13

(i)sin θ

$\sin\theta=\frac{Opposite}{Hypotenuse}=\frac{12}{13}$

(ii)cos θ

$\cos\theta=\frac{Adjacent}{Hypotenuse}=\frac{5}{13}$

(iii)tan θ

$\tan\theta=\frac{Opposite}{Adjacent}=\frac{12}{5}$

(iv)cosec θ

$\cosec\theta=\frac{1}{\sin\theta}=\frac{13}{12}$

(v)sec θ

$\sec\theta=\frac{1}{\cos\theta}=\frac{13}{5}$

(vi)cot θ

$\cot\theta=\frac{1}{\tan\theta}=\frac{5}{12}$