Depending on in which quadrant an angle `theta` lies, functions `costheta` and `sintheta` may be positive or negative. In the second column of the given lable are specified whether these functions are positive or negative and in the first column are specified quadrants.
`{:("Column-I","Column-II"),((A)" First",(P)sintheta" is positive"),((B)" Second",(Q)sintheta" is negative"),((C)" Third",(R)costheta" is negative"),((D)" Fourth",(S)tantheta" and "sintheta"both are negative"),(,(T)sectheta" is negative and "sintheta" is positive"):}`

Answer

To solve the problem, we need to determine the signs of the trigonometric functions \( \sin \theta \), \( \cos \theta \), and \( \tan \theta \) in each of the four quadrants and match them with the given options in Column II.

### Step 1: Analyze the First Quadrant
In the **First Quadrant** (0° to 90°):
- \( \sin \theta \) is positive
- \( \cos \theta \) is positive
- \( \tan \theta \) is positive

**Matching:**
- \( \sin \theta \) is positive (P) → Matches with A.

### Step 2: Analyze the Second Quadrant
In the **Second Quadrant** (90° to 180°):
- \( \sin \theta \) is positive
- \( \cos \theta \) is negative
- \( \tan \theta \) is negative

**Matching:**
- \( \sin \theta \) is positive (P) → Matches with B.
- \( \cos \theta \) is negative (R) → Matches with B.
- \( \sec \theta \) is negative and \( \sin \theta \) is positive (T) → Matches with B.

### Step 3: Analyze the Third Quadrant
In the **Third Quadrant** (180° to 270°):
- \( \sin \theta \) is negative
- \( \cos \theta \) is negative
- \( \tan \theta \) is positive

**Matching:**
- \( \sin \theta \) is negative (Q) → Matches with C.
- \( \cos \theta \) is negative (R) → Matches with C.
- \( \tan \theta \) is positive, so \( \tan \theta \) and \( \sin \theta \) both are negative (S) → Does not match.

### Step 4: Analyze the Fourth Quadrant
In the **Fourth Quadrant** (270° to 360°):
- \( \sin \theta \) is negative
- \( \cos \theta \) is positive
- \( \tan \theta \) is negative

**Matching:**
- \( \sin \theta \) is negative (Q) → Matches with D.
- \( \tan \theta \) and \( \sin \theta \) both are negative (S) → Matches with D.
- \( \sec \theta \) is positive, so it does not match.

### Final Matching:
- A → P (First Quadrant)
- B → P, R, T (Second Quadrant)
- C → Q, R (Third Quadrant)
- D → Q, S (Fourth Quadrant)

### Summary of Matches:
- A → P
- B → P, R, T
- C → Q, R
- D → Q, S