Question

Sketch a right triangle corresponding to the trigonometric function of the acute angle 8. Then find the exact values of the other five trigonometric functions of

Answer 1

Given:

`cot(\theta)=2`

Using the trig ratio of cot;

`\begin{gathered} cot\theta=(1)/(tan\theta) \\ tan\theta=(opposite)/(adjacent) \\ Hence, \\ cot\theta=(adjacent)/(opposite) \\ cot\theta=(2)/(1) \\ adjacent=2 \\ opposite=1 \end{gathered}`

The hypotenuse is gotten using the Pythagoras theorem;

`\begin{gathered} h^2=2^2+1^2 \\ h^2=4+1 \\ h^2=5 \\ h=√(5) \end{gathered}`

Thus, the sketch of the right triangle is;

`\begin{gathered} Thus, \\ opposite=1 \\ adjacent=2 \\ hypotenuse=√(5) \end{gathered}`

Hence,

`\begin{gathered} sin\theta=(opposite)/(hypotenuse) \\ sin\theta=(1)/(√(5)) \\ sin\theta=(1)/(√(5))*(√(5))/(√(5))=(√(5))/(5) \\ sin\theta=(√(5))/(5) \end{gathered}`
`\begin{gathered} cos\theta=(adjacent)/(hypotenuse) \\ cos\theta=(2)/(√(5)) \\ cos\theta=(2√(5))/(5) \end{gathered}`
`\begin{gathered} tan\theta=(opposite)/(adjacent) \\ tan\theta=(1)/(2) \end{gathered}`
`\begin{gathered} csc\theta=(1)/(sin\theta) \\ csc\theta=(1)/((√(5))/(5)) \\ csc\theta=√(5) \end{gathered}`
`\begin{gathered} sec\theta=(1)/(cos\theta) \\ sec\theta=(1)/((2√(5))/(5)) \\ sec\theta=(√(5))/(2) \end{gathered}`