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.
[tex]\begin{gathered} Hypotenuse^2=Altitude^2+Base^2 \\ |MT|^2=|MA|^2+|AT|^2 \end{gathered}[/tex]
Substitute the known values:
[tex]\begin{gathered} |MT|^2=12^2+5^2 \\ \lvert MT\rvert=\sqrt{12^2+5^2}=\sqrt{169}=13 \end{gathered}[/tex]
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
[tex]\sin\theta=\frac{Opposite}{Hypotenuse}=\frac{12}{13}[/tex]
(ii)cos θ
[tex]\cos\theta=\frac{Adjacent}{Hypotenuse}=\frac{5}{13}[/tex]
(iii)tan θ
[tex]\tan\theta=\frac{Opposite}{Adjacent}=\frac{12}{5}[/tex]
(iv)cosec θ
[tex]\cosec\theta=\frac{1}{\sin\theta}=\frac{13}{12}[/tex]
(v)sec θ
[tex]\sec\theta=\frac{1}{\cos\theta}=\frac{13}{5}[/tex]
(vi)cot θ
[tex]\cot\theta=\frac{1}{\tan\theta}=\frac{5}{12}[/tex]
