Find the quadrant containing 𝜃 if the given conditions are true.
(a)    cos 𝜃 < 0 and sin 𝜃 > 0

quadrant I
quadrant II
quadrant III
quadrant IV

(b)    sin 𝜃 < 0 and cot 𝜃 > 0

quadrant I
quadrant II
quadrant III
quadrant IV

(c)    csc 𝜃 > 0 and sec 𝜃 > 0

quadrant I
quadrant II
quadrant III
quadrant IV

(d)    sec 𝜃 > 0 and tan 𝜃 < 0

quadrant I
quadrant II
quadrant III
quadrant IV

To determine the quadrant containing \( \theta \) based on the given conditions, we can analyze each case using the signs of the trigonometric functions in each quadrant.

### (a) \( \cos \theta < 0 \) and \( \sin \theta > 0 \)

- **Quadrant II**: In this quadrant, cosine is negative and sine is positive.

**Answer**: **Quadrant II**

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### (b) \( \sin \theta < 0 \) and \( \cot \theta > 0 \)

- **Quadrant IV**: In this quadrant, sine is negative and cotangent (which is positive when both sine and cosine are either both positive or both negative) indicates cosine is also negative.

**Answer**: **Quadrant IV**

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### (c) \( \csc \theta > 0 \) and \( \sec \theta > 0 \)

- **Quadrant I**: In this quadrant, both cosecant (which is the reciprocal of sine) and secant (which is the reciprocal of cosine) are positive.

**Answer**: **Quadrant I**

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### (d) \( \sec \theta > 0 \) and \( \tan \theta < 0 \)

- **Quadrant IV**: In this quadrant, secant is positive (since cosine is positive) and tangent is negative (since sine is negative).

**Answer**: **Quadrant IV**

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### Summary of Answers:
- (a) Quadrant II
- (b) Quadrant IV
- (c) Quadrant I
- (d) Quadrant IV

Would you like more details or have any questions about these results? Here are some related questions for further exploration:

1. What are the signs of sine and cosine in each quadrant?
2. How can you remember the signs of trigonometric functions in different quadrants?
3. What are the definitions of cotangent, cosecant, and secant?
4. Can you provide a unit circle illustration to visualize these quadrants?
5. How do you determine the reference angle for a given \( \theta \)? 

**Tip**: Familiarize yourself with the unit circle; it’s a great visual aid for understanding the signs of trigonometric functions in different quadrants!

Find the quadrant containing 𝜃 if the given conditions are true: (a) cos 𝜃 < 0 and sin 𝜃 > 0; (b) sin 𝜃 < 0 and cot 𝜃 > 0; (c) csc 𝜃 > 0 and sec 𝜃 > 0; (d) sec 𝜃 > 0 and tan 𝜃 < 0.

Explore how to determine the quadrant of θ based on trigonometric conditions including cos θ < 0, sin θ > 0, and others. This guide covers key trigonometric concepts and signs in different quadrants. Ideal for high school students.

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