https://file.aiquickdraw.com/mathbot/file/1729785848892s6usntcf.jpg

Let's go through each question one by one:

### 1. (4 pts) Name the quadrant in which the angle lies, and its reference angle.

You are given two angles: 
- \( 600^\circ \)
- \( \frac{-5\pi}{3} \)

#### (a) \( 600^\circ \)
First, find the equivalent angle between \( 0^\circ \) and \( 360^\circ \) by subtracting multiples of \( 360^\circ \):
\[
600^\circ - 360^\circ = 240^\circ
\]
Now, \( 240^\circ \) lies in Quadrant III. The reference angle is the angle made with the x-axis, so:
\[
360^\circ - 240^\circ = 120^\circ \quad \text{Reference angle} = 120^\circ
\]

#### (b) \( \frac{-5\pi}{3} \)
Convert the negative angle to a positive one by adding \( 2\pi \):
\[
\frac{-5\pi}{3} + 2\pi = \frac{-5\pi + 6\pi}{3} = \frac{\pi}{3}
\]
So, \( \frac{\pi}{3} \) lies in **Quadrant I**. The reference angle is simply \( \frac{\pi}{3} \).

### 2. (10 pts) Find each of the following.

#### (a) Find an angle coterminal to \( \frac{-3\pi}{4} \) that lies between \( 0 \) and \( 2\pi \).
To make this angle positive, add \( 2\pi \):
\[
\frac{-3\pi}{4} + 2\pi = \frac{-3\pi + 8\pi}{4} = \frac{5\pi}{4}
\]
So, a coterminal angle between 0 and \( 2\pi \) is \( \frac{5\pi}{4} \).

#### (b) Determine cosine/sine in different quadrants.

You are asked to find four positive angles with the same reference angle as \( \frac{-3\pi}{4} \), which is \( \frac{\pi}{4} \), and determine if they have the same cosine or sine.

1. **Angle in QI**: The reference angle is \( \frac{\pi}{4} \). 
   - \( \cos(\frac{-3\pi}{4}) = -\frac{1}{\sqrt{2}} \)
   - In QI, \( \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \) (cosine is positive in QI).
   - **Same cosine? No**

2. **Angle in QII**: The angle is \( \pi - \frac{\pi}{4} = \frac{3\pi}{4} \).
   - In QII, cosine is negative, so \( \cos(\frac{3\pi}{4}) = -\frac{1}{\sqrt{2}} \).
   - **Same cosine? Yes**

3. **Angle in QIII**: The angle is \( \pi + \frac{\pi}{4} = \frac{5\pi}{4} \).
   - In QIII, sine is negative, so \( \sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}} \).
   - **Same sine? Yes**

4. **Angle in QIV**: The angle is \( 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \).
   - In QIV, sine is negative, so \( \sin(\frac{7\pi}{4}) = -\frac{1}{\sqrt{2}} \).
   - **Same sine? Yes**

### 3. (6 pts) Find the values of the given trig functions without using a calculator.

#### (a) \( \cos\left(\frac{-5\pi}{3}\right) \)
We found earlier that \( \frac{-5\pi}{3} \) is coterminal with \( \frac{\pi}{3} \). Thus:
\[
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\]
Therefore:
\[
\cos\left(\frac{-5\pi}{3}\right) = \frac{1}{2}
\]

#### (b) \( \tan\left(\frac{-5\pi}{3}\right) \)
Using the same coterminal angle \( \frac{\pi}{3} \):
\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\]
Since the original angle is negative, \( \tan\left(\frac{-5\pi}{3}\right) = -\sqrt{3} \).

---

Let me know if you'd like more details or have any further questions! Here are five related questions you might want to explore:

1. What is the formula for finding coterminal angles?
2. How do you find reference angles in different quadrants?
3. Can you explain why the signs of trigonometric functions change in different quadrants?
4. What is the geometric interpretation of sine and cosine on the unit circle?
5. How can you quickly identify which quadrant an angle belongs to?

**Tip**: When working with trigonometric functions, always check the quadrant to determine the correct sign of the value.

Find the quadrant, reference angle, coterminal angles, and trigonometric values for the given angles. Determine cosine/sine for angles in different quadrants.

Solve questions involving quadrant identification, reference angles, and finding coterminal angles for angles like 600°, -5π/3, and -3π/4. Learn how cosine and sine vary across different quadrants with detailed solutions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation