Question

Name the quadrant in which the angle \(\theta\) lies. $$ \cos \theta>0, \quad \cot \theta<0 $$

Short answer

Fourth quadrant

Step-by-step solution

- Understand the given conditions

The conditions given are \( \cos \theta > 0 \) and \( \cot \theta < 0 \). The cosine of an angle is positive in the first and fourth quadrants, while the cotangent of an angle is negative in the second and fourth quadrants.

- Analyze compatibility of conditions

Since \( \cos \theta \) is positive only in the first and fourth quadrants, and \( \cot \theta \) is negative only in the second and fourth quadrants, identify the quadrant that fits both conditions.

- Identify the common quadrant

The fourth quadrant is the only quadrant where both conditions \( \cos \theta > 0 \) and \( \cot \theta < 0 \) are satisfied.

Key concepts

Cosine Function

The cosine function, often denoted as \( \cos \theta \), is one of the primary functions in trigonometry. It measures the horizontal distance of a point on the unit circle from the origin. Here's what you need to know about the cosine function:

• When the angle \( \theta \) is in the first quadrant (0° to 90°), \( \cos \theta \) is positive.
• In the second quadrant (90° to 180°), \( \cos \theta \) turns negative.
• In the third quadrant (180° to 270°), \( \cos \theta \) remains negative.
• Finally, in the fourth quadrant (270° to 360° or -90° to 0°), \( \cos \theta \) becomes positive again.

Remember, cosine helps in determining the position of an angle with respect to the x-axis. It is essential to understanding many trigonometric identities and applications.

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is another crucial trigonometric function. It is defined as the ratio of the cosine of an angle to the sine of that angle: \[ \cot \theta = \frac{ \cos \theta }{ \sin \theta } \] Some key points about the cotangent function:

• In the first quadrant, both sine and cosine are positive, so \( \cot \theta \) is positive.
• In the second quadrant, sine is positive but cosine is negative, so \ \( \cot \theta \) is negative.
• In the third quadrant, both sine and cosine are negative, making \ \( \cot \theta \) positive.
• In the fourth quadrant, sine is negative while cosine is positive, therefore \ \( \cot \theta \) is negative.

The sign of \( \cot \theta \) depends on the signs of both sine and cosine, which vary based on the quadrant of the angle.

Angle Quadrants

Understanding which quadrant an angle lies in is fundamental in trigonometry. The unit circle is divided into four quadrants:

• First Quadrant: The angle ranges from 0° to 90°. Here, both sine and cosine are positive.
• Second Quadrant: The angle ranges from 90° to 180°. In this quadrant, sine is positive, but cosine is negative.
• Third Quadrant: The angle ranges from 180° to 270°. Both sine and cosine are negative in this quadrant.
• Fourth Quadrant: The angle ranges from 270° to 360° (or from -90° to 0°). Here, sine is negative, but cosine is positive.

When given specific conditions like \( \cos \theta > 0 \) and \( \cot \theta < 0 \), it helps to know the signs of the trigonometric functions in each quadrant. As we saw in the problem, \ \( \cos \theta \) is positive only in the first and fourth quadrants, but \ \( \cot \theta \) is negative only in the second and fourth quadrants. Hence, the angle \ \( \theta \) must lie in the fourth quadrant since it's the only quadrant satisfying both conditions.