Question

Find the exact values of the six trigonometric functions of the angle 0 shown in the future

Answer 1

From the given right-angle triangle, we are provided with the following sides;

`\begin{gathered} Hypotenuse=17 \\ Opposite=8 \end{gathered}`

We will make use of the Pythagoras theorem to obtain the third side, which is the Adjacent side.

Thus, we have:

`\begin{gathered} H^2=O^2+A^2 \\ 17^2=8^2+A^2 \\ 289=64+A^2 \\ 289-64=A^2 \\ 225=A^2 \\ A=\sqrt[]{225} \\ A=15 \end{gathered}`

i)

`\begin{gathered} Sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}} \\ Sin\theta=(8)/(17) \end{gathered}`

ii)

`\begin{gathered} Cos\theta=\frac{Adjacent}{\text{Hypotenuse}} \\ \text{Cos}\theta=(15)/(17) \end{gathered}`

iii)

`\begin{gathered} \text{Tan}\theta=\frac{\text{Opposite}}{\text{Adjacent}} \\ \text{Tan}\theta=(8)/(15) \end{gathered}`

iv)

`\begin{gathered} co\sec (\csc )\text{ is the inverse/reciprocal of sine} \\ \text{csc }\theta=\frac{Hypotenuse}{\text{Opposite}} \\ \csc \theta=(17)/(8) \\ \end{gathered}`

v)

`\begin{gathered} \sec \theta\text{ is the inverse/reciprocal of cos }\theta \\ \sec \theta=\frac{\text{Hypotenuse}}{\text{Adjacent}} \\ \sec \theta=(17)/(15) \end{gathered}`

vi)

`\begin{gathered} \cot \theta\text{ is the inverse/reciprocal of tan }\theta \\ \cot \theta=\frac{\text{Adjacent }}{\text{Opposite}} \\ \cot \theta=(15)/(8) \end{gathered}`