Chapter 8, Problem 2RE

In Problems 1 and 2, find the exact value of the six trigonometric functions of the angle $\theta$ in each figure.

To determine

To find: The exact value of the six trigonometric functions of the angle $\text{u}$ in each figure:

Answer to Problem 2RE

Solution:

$\text{sin}\theta = \frac{\sqrt{3}}{2}$

$\text{cos}\theta = \frac{1}{2}$

$\text{tan}\theta = \frac{\sqrt{3}}{2}$

$\text{cosec}\theta = \frac{2}{\sqrt{3}}$

$\text{sec}\theta = 2$

$\text{cot}\theta = \frac{1}{\sqrt{3}}$

Explanation of Solution

Given:

Formula used:

Pythagorean Theorem

$\text{sin}\theta = \frac{opposite side}{hypotenuse}$

$\text{cos}\theta = \frac{adjacent side}{hypotenuse}$

$\text{tan}\theta = \frac{\text{sin}\theta }{\text{cos}\theta } = \frac{opposite side}{adjacent side}$

$\text{cosec}\theta = \frac{1}{\text{sin}\theta } = \frac{hypotenuse}{opposite side}$

$\text{sec}\theta = \frac{1}{\text{cos}\theta } = \frac{hypotenuse}{adjacent side}$

$\text{cot}\theta = \frac{\text{cos}\theta }{\text{sin}\theta } = \frac{adjacent side}{opposite side}$

Calculation:

$\begin{array}{l}{2}^2 + {\text{opp}}^2 = 4^2\\ 16 - 4 = {\text{opp}}^2\end{array}$

${\text{Opposite}}^2 = {(2\surd 3)}^2$

So removing square on both sides we will get

$\text{Opposite} = 2\surd 3$

Ratios:

$\text{Opposite} \text{side} = 2\surd 3$

$\begin{array}{l}\text{Adjacent} \text{side} = 2\\ \text{Hypotenuse} = 4\end{array}$

$\text{sin}\theta = \frac{opposite side}{hypotenuse} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}$

$\text{cos}\theta = \frac{adjacent side}{hypotenuse} = \frac{2}{4} = \frac{1}{2}$

$\text{tan}\theta = \frac{\text{sin}\theta }{\text{cos}\theta } = \frac{opposite side}{adjacent side} = \surd 3$

$\text{cosec}\theta = \frac{1}{\text{sin}\theta } = \frac{hypotenuse}{opposite side} = \frac{2}{\sqrt{3}}$

$\text{sec}\theta = \frac{1}{\text{cos}\theta } = \frac{hypotenuse}{adjacent side} = 2$

$\text{cot}\theta = \frac{\text{cos}\theta }{\text{sin}\theta } = \frac{adjacent side}{opposite side} = \frac{1}{\sqrt{3}}$