Chapter 6, Problem 8RCC

To determine

To find: the reference angle using angle in standard position.

Answer to Problem 8RCC

The reference angle is the difference between the angle and the closest multiple of ${180}^0$ .

Explanation of Solution

Given:

  $\text{θ}$ is angle in standard position.

Concept used:

Reference angle:

the reference angle is the difference between the angle and the closest multiple of ${180}^0$ .

In the form of $\text{θ=π-θ}$ , $\text{θ=π+θ}$ , $\text{θ=2π+θ}$ and $\text{θ=2π-θ}$ .

Calculation:

Assuming that an angle in the standard position, its reference angle is the smallest angle it can make between the terminal ray and the whole left-right x-axis.

Mathematically, the reference angle is the difference between the angle and the closest multiple of ${180}^0$ .

Example:

  ${\text{θ=-245}}^{\text{0}}.$

An angle ${\text{θ=-245}}^{\text{0}}.$ lies in the 2^nd quadrant.

Thus the closest multiple of ${\text{180}}^{\text{0}}$ $\text{to}$ $\text{θ}$ is ${\text{-180}}^0$ .

The difference between these two is

  $\left|-{245}^0-\left(-{180}^0\right)\right|$ .

  $\Rightarrow \left|-{245}^0+{180}^0\right|$ .

  $\Rightarrow \left|-{65}^0\right|$ .

  ${65}^0$ .