Which one is incorrect?

Options

A. `sintheta=(3)/(4)`
B. `costheta=(1)/(3)`
C. `sectheta=(1)/(2)`
D. `tantheta=9`

Answer

To determine which option is incorrect, we will analyze each option step by step based on the properties of trigonometric functions.

### Step 1: Analyze Option 1
**Option 1:** \( \sin \theta = \frac{3}{4} \)

- The range of the sine function is from -1 to 1.
- \( \frac{3}{4} = 0.75 \), which lies within the range of sine.

**Conclusion:** Option 1 is correct.

### Step 2: Analyze Option 2
**Option 2:** \( \cos \theta = \frac{1}{3} \)

- The range of the cosine function is also from -1 to 1.
- \( \frac{1}{3} \approx 0.333 \), which lies within the range of cosine.

**Conclusion:** Option 2 is correct.

### Step 3: Analyze Option 3
**Option 3:** \( \sec \theta = \frac{1}{2} \)

- The secant function is defined as \( \sec \theta = \frac{1}{\cos \theta} \).
- The range of secant is \( (-\infty, -1] \cup [1, \infty) \).
- Since \( \frac{1}{2} = 0.5 \), it does not lie within the range of secant.

**Conclusion:** Option 3 is incorrect.

### Step 4: Analyze Option 4
**Option 4:** \( \tan \theta = 9 \)

- The range of the tangent function is from \( -\infty \) to \( \infty \).
- Since 9 is a real number, it lies within the range of tangent.

**Conclusion:** Option 4 is correct.

### Final Conclusion
The incorrect option is **Option 3**: \( \sec \theta = \frac{1}{2} \).

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