Question

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

Short answer

The third quadrant (Quadrant III).

Step-by-step solution

Understand the Problem

Identify the given conditions: \ \ 1) \ \ \( \sec \theta<0 \ \ \) - The secant function is negative when \( \cos \theta \) is negative because \( \sec \theta \) is the reciprocal of \( \cos \theta \), so \( \cos \theta<0 \).\ 2) \ \ \( \tan \theta>0 \ \ \) - The tangent function is positive.

Identify Where \( \cos \theta \) is Negative

The cosine function \( \cos \theta \) is negative in the second quadrant (Quadrant II) and the third quadrant (Quadrant III).

Identify Where \( \tan \theta \) is Positive

The tangent function \( \tan \theta \) is positive in the first quadrant (Quadrant I) and the third quadrant (Quadrant III).

Determine the Common Quadrant

Combine both conditions to find the common quadrant: \ \ - Only the third quadrant (Quadrant III) satisfies both \( \cos \theta<0 \) and \( \tan \theta>0 \).

Conclusion

The angle \( \theta \) lies in the third quadrant (Quadrant III).

Key concepts

Secant Function

The secant function, abbreviated as \( \sec \theta \), is one of the important trigonometric functions. It is defined as the reciprocal of the cosine function. Mathematically, this means:
\[ \sec \theta = \frac{1}{\cos \theta} \]
Understanding the sign (positive or negative) of \( \sec \theta \) is directly tied to knowing the behavior of \( \cos \theta \). Since \( \sec \theta \) is simply the inverse of \( \cos \theta \), when \( \cos \theta \) is negative, \( \sec \theta \) will also be negative. Conversely, when \( \cos \theta \) is positive, \( \sec \theta \) will be positive too.
In different quadrants on the unit circle:

• In Quadrant I: \( \sec \theta \) is positive because \( \cos \theta \) is positive.
• In Quadrant II: \( \sec \theta \) is negative because \( \cos \theta \) is negative.
• In Quadrant III: \( \sec \theta \) is negative because \( \cos \theta \) is negative.
• In Quadrant IV: \( \sec \theta \) is positive because \( \cos \theta \) is positive.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is another fundamental trigonometric function. It is defined as the ratio of the sine and cosine functions:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
The sign of \( \tan \theta \) will depend on the signs of \( \sin \theta \) and \( \cos \theta \). The tangent function helps in understanding the slope of an angle and has useful properties in trigonometry.
In different quadrants:

• In Quadrant I: Both \( \sin \theta \) and \( \cos \theta \) are positive, so \( \tan \theta \) is positive.
• In Quadrant II: \( \sin \theta \) is positive, but \( \cos \theta \) is negative, making \( \tan \theta \) negative.
• In Quadrant III: Both \( \sin \theta \) and \( \cos \theta \) are negative, making \( \tan \theta \) positive.
• In Quadrant IV: \( \sin \theta \) is negative, while \( \cos \theta \) is positive, making \( \tan \theta \) negative.

Knowing when \( \tan \theta \) is positive or negative helps in identifying the correct quadrant for an angle.

Quadrants

The coordinate plane is divided into four quadrants, each of which has distinct properties related to the signs of the trigonometric functions. The quadrants are:

• Quadrant I: Where both sine and cosine are positive, making secant and tangent both positive.
• Quadrant II: Sine is positive and cosine is negative, making secant negative and tangent negative.
• Quadrant III: Both sine and cosine are negative, making secant negative and tangent positive.
• Quadrant IV: Sine is negative and cosine is positive, making secant positive and tangent negative.

The quadrant in which an angle lies is important for understanding the behavior and sign of the trigonometric functions associated with that angle. For example, in this exercise: Given \( \sec \theta < 0 \) and \( \tan \theta > 0 \), the angle \( \theta \) must lie in Quadrant III, where secant is negative and tangent is positive.