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Chapter 1: Problem 6

Draw the angle using a ray through the origin, and determine whether the sine, cosine, and tangent of that angle are positive, negative, zero, or undefined. $$\frac{4 \pi}{3}$$

Short Answer

Expert verified
Sine is negative, cosine is negative, and tangent is positive.

Step by step solution

01

Identify the Quadrant of the Angle

To identify the quadrant, first convert the given angle \( \frac{4\pi}{3} \) from radians to a common form by dividing \( 4\pi \) by \( 3 \). Notice that \( \pi \) radians equal \( 180 \) degrees, so \( \frac{4\pi}{3} \approx 240 \) degrees. This angle is in the third quadrant.
02

Determine the Sine of the Angle

In the third quadrant, both sine and cosine values are negative and tangent is positive. To determine whether the sine of \( \frac{4\pi}{3} \) is positive, negative, or zero: since the angle is in the third quadrant, \( \sin \left( \frac{4\pi}{3} \right) < 0 \), hence sine is negative.
03

Determine the Cosine of the Angle

Similar to the sine function, check the cosine: since the angle \( \frac{4\pi}{3} \) is still in the third quadrant, \( \cos \left( \frac{4\pi}{3} \right) < 0 \), indicating that the cosine is negative as well.
04

Determine the Tangent of the Angle

For the tangent function: in the third quadrant both sine and cosine are negative, making tangent positive, because \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) becomes positive when both numerator and denominator (sine and cosine here) are negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radians to Degrees Conversion
Converting radians to degrees is an essential skill in trigonometry. Angles can be expressed in either radians or degrees, and understanding how to transition between these units can simplify many trigonometric problems. The conversion formula between radians and degrees is as follows:
\[ ext{Degrees} = ext{Radians} imes \left(\frac{180}{\pi}\right) \]
For example, with the angle \( \frac{4\pi}{3} \) radians, we can apply the formula:
\[\frac{4\pi}{3} \times \left(\frac{180}{\pi}\right) = 240 \text{ degrees}\]
This conversion reveals that \( \frac{4\pi}{3} \) radians is equivalent to 240 degrees, an angle measurement often more familiar to students.
  • Remember: \( \pi \) radians equals 180 degrees, a critical relationship in trigonometry.
  • Conversion is a linear transformation: simply multiply by \( \frac{180}{\pi} \).
Quadrant Analysis
Quadrant analysis is key to understanding the sign of trigonometric functions. The unit circle is divided into four quadrants, with each quadrant affecting the sign of sine, cosine, and tangent differently. The angles in each quadrant follow these simple rules:
  • First Quadrant: All trigonometric functions are positive.
  • Second Quadrant: Sine is positive, while cosine and tangent are negative.
  • Third Quadrant: Both sine and cosine are negative, but tangent is positive.
  • Fourth Quadrant: Cosine is positive, while sine and tangent are negative.
For the angle \( 240 \) degrees (or \( \frac{4\pi}{3} \) radians):
- Locate it in the third quadrant.
- In this quadrant, due to both sine and cosine being negative, the tangent remains positive.
Simply determining the quadrant can help predict the nature (positive, negative, or zero) of these functions even before calculation.
Sine Cosine Tangent Signs
Recognizing the signs of sine, cosine, and tangent in different quadrants directs you towards the correct evaluation of a trigonometric function. These signs are consistent throughout the quadrants and can be summarized based on their position on the unit circle:
- **Sine** refers to the y-value on the unit circle.
- **Cosine** refers to the x-value on the unit circle.
- **Tangent** is the ratio of sine to cosine, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
  • In the **third quadrant**, both sine and cosine are negative, making their ratio (tangent) positive.
  • This means for \( \frac{4\pi}{3} \):
    - \( \sin\left(\frac{4\pi}{3}\right) < 0 \)
    - \( \cos\left(\frac{4\pi}{3}\right) < 0 \)
    - \( \tan\left(\frac{4\pi}{3}\right) > 0 \)
Use quadrant analysis paired with the signs of sine, cosine, and tangent to correctly solve trigonometric problems across various scenarios.

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