Question

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

Answer 1

• 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=√(12^2+5^2)=√(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=(Opposite)/(Hypotenuse)=(12)/(13)`

(ii)cos θ

`\cos\theta=(Adjacent)/(Hypotenuse)=(5)/(13)`

(iii)tan θ

`\tan\theta=(Opposite)/(Adjacent)=(12)/(5)`

(iv)cosec θ

`\cosec\theta=(1)/(\sin\theta)=(13)/(12)`

(v)sec θ

`\sec\theta=(1)/(\cos\theta)=(13)/(5)`

(vi)cot θ

`\cot\theta=(1)/(\tan\theta)=(5)/(12)`