## How to Sketch Angles in Standard Position

### Sketch Each Angle in Standard Position

A standard-position angle has its vertex at the plane's origin. Along the positive $$x$$-axis is where its initial ray (beginning side) is located. From the beginning side, its terminal ray (finishing side) travels counterclockwise.

The angle is negative if the terminal ray rotates in a clockwise direction. Greek letters are often used to identify angles in their standard position.

### How to Make a Negative Angle in Standard Position

Drawing a picture will help you figure out how to solve a problem in trigonometry many times, if not permanently. So what should you do if someone asks you to draw an angle larger than $$360$$ degrees? Or a negative angle? Don't worry, the steps below will assist you:

Let's solve an example step by step

Example: Draw a $$-640°$$ angle.

• By adding $$360$$ degrees, get the co-terminal angle.
$$-640° \ + \ 360° \ = \ -280°$$
• If the angle is still negative, keep adding $$360$$ degrees until bring you a positive angle in the standard position.
$$-280° \ + \ 360° \ = \ 80°$$
• Draw the angle you get in Step $$2$$.

### There are two things you should notice:

• Do you move in a clockwise or counterclockwise direction?
• How many times do you circle the coordinate plane's origin?

Since you are determining a negative angle, this angle begins at $$0$$ on the $$x$$-axis and travels clockwise.

### Summary

- The standard position of an angle is when its vertex is located at the origin and its initial side extends along the positive $$x$$-axis.
- A positive angle is the angle measured in a counterclockwise direction from the initial side to the terminal side.
- A negative angle is the angle measured in a clockwise direction from the initial side to the terminal side.

### Exercises for Sketch Each Angle in Standard Position

1) Draw the angle in standard position: $$α \ = \ -320°$$

2) Draw the angle in standard position: $$α \ = \ -490°$$

3) Draw the angle in standard position: $$α \ = \ \frac{31π}{18}$$

4) Draw the angle in standard position: $$α \ = \ \frac{11π}{6}$$

5) Draw the angle in standard position: $$α \ = \ -\frac{38π}{9}$$

6) Draw the angle in standard position: $$α \ = \ -\frac{55π}{18}$$

7) Draw the angle in standard position: $$α \ = \ -\frac{23π}{9}$$

8) Draw the angle in standard position: $$α \ = \ - 440°$$

9) Draw the angle in standard position: $$α \ = \ - 390°$$

10) Draw the angle in standard position: $$α \ = \ -\frac{5π}{2}$$

1) Draw the angle in standard position: $$α \ = \ -320°$$

$$\color{red}{-320° \ + \ 360° \ = \ 40°}$$

2) Draw the angle in standard position: $$α \ = \ -490°$$

$$\color{red}{-490° \ + \ 360° \ = \ 130°}$$

3) Draw the angle in standard position: $$α \ = \ \frac{31π}{18}$$

$$\color{red}{\frac{31π}{18} \ \times \ \frac{180}{π} \ = \ 310°}$$

4) Draw the angle in standard position: $$α \ = \ \frac{11π}{6}$$

$$\color{red}{\frac{11π}{6} \ \times \ \frac{180}{π} \ = \ 330°}$$

5) Draw the angle in standard position: $$α \ = \ -\frac{38π}{9}$$

$$\color{red}{-\frac{38π}{9} \ \times \ \frac{180}{π} \ = \ -760°}$$
$$\color{red}{-760° \ + \ 360° \ = \ -400°}$$
$$\color{red}{-400° \ + \ 360° \ = \ -40°}$$
$$\color{red}{-40° \ + \ 360° \ = \ 320°}$$

6) Draw the angle in standard position: $$α \ = \ -\frac{55π}{18}$$

$$\color{red}{-\frac{55π}{18} \ \times \ \frac{180}{π} \ = \ -550°}$$
$$\color{red}{-550° \ + \ 360° \ = \ -190°}$$
$$\color{red}{-190° \ + \ 360° \ = \ 170°}$$

7) Draw the angle in standard position: $$α \ = \ -\frac{23π}{9}$$

$$\color{red}{-\frac{23π}{9} \ \times \ \frac{180}{π} \ = \ -460°}$$
$$\color{red}{-460° \ + \ 360° \ = \ -100°}$$
$$\color{red}{-100° \ + \ 360° \ = \ 260°}$$

8) Draw the angle in standard position: $$α \ = \ - 440°$$

$$\color{red}{-440° \ + \ 360° \ = \ -80°}$$
$$\color{red}{-80° \ + \ 360° \ = \ 280°}$$

9) Draw the angle in standard position: $$α \ = \ - 390°$$

$$\color{red}{-390° \ + \ 360° \ = \ -30°}$$
$$\color{red}{-30° \ + \ 360° \ = \ 330°}$$

10) Draw the angle in standard position: $$α \ = \ -\frac{5π}{2}$$

$$\color{red}{-\frac{5π}{2} \ \times \ \frac{180}{π} \ = \ -450°}$$
$$\color{red}{-450° \ + \ 360° \ = \ -90°}$$
$$\color{red}{-90° \ + \ 360° \ = \ 270°}$$

## Sketch Each Angle in Standard Position Practice Quiz

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