[tex]$\cos\theta=\frac{\text{5}}{\text{13}}, \ \sin\theta=\frac{\text{12}}{13}, \ \tan\theta=\frac{\text{12}}{\text{5}}[/tex]
[tex]$\sec\theta=\frac{\text{13}}{\text{5}},\ \csc \theta=\frac{\text{13}}{\text{12}} , \ \cot \theta=\frac{\text{5}}{\text{12}}[/tex]
Solution:
Given triangle is a right triangle.
Base = 5, Hypotenuse = 13, Height = ?
Height (h) = opposite
In right triangle, square of the hypotenuse is equal to the sum of the squares of the other two sides.
[tex]5^2+h^2=13^2[/tex]
[tex]25+h^2=169[/tex]
[tex]h^2=144[/tex]
Taking square root on both sides.
h = 12
[tex]$\cos\theta=\frac{\text{Base}}{\text{Hypotenuse}}[/tex]
[tex]$\cos\theta=\frac{\text{5}}{\text{13}}[/tex]
[tex]$\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}[/tex]
[tex]$\sin\theta=\frac{\text{12}}{13}[/tex]
[tex]$\tan\theta=\frac{\text{Opposite}}{\text{Base}}[/tex]
[tex]$\tan\theta=\frac{\text{12}}{\text{5}}[/tex]
[tex]$\sec\theta=\frac{\text{Hypotenuse}}{\text{Base }}[/tex]
[tex]$\sec\theta=\frac{\text{13}}{\text{5}}[/tex]
[tex]$\csc \theta=\frac{\text{Hypotenuse}}{\text{Opposite }}[/tex]
[tex]$\csc \theta=\frac{\text{13}}{\text{12}}[/tex]
[tex]$\cot \theta=\frac{\text{Base }}{\text{Opposite}}[/tex]
[tex]$\cot \theta=\frac{\text{5}}{\text{12}}[/tex]
