## How to Sketch Angles in Standard Position

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### 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

### More Trigonometric Functions courses

- How to Find the Length of Arc and the Area of Sector
- How to Write Each Measure in Radians
- How to Find Missing Sides and Angles of a Right Triangle
- How to Write Each Measure in Degrees
- How to Evaluate Each Trigonometric Function
- How to Sketch Angles in Standard Position
- How to Find Co-terminal Angles and Reference Angles
- How to Find Trigonometric Ratios of General Angles