Which of the following is possible ?

Options

A. `cos theta =7/5`
B. `sin theta =8/5`
C. `sec theta =4/5`
D. `tan theta =45`

Answer

To determine which of the given expressions is possible, we will analyze each option based on the properties of trigonometric functions.

### Step-by-Step Solution:

1. **Analyzing the first expression: \( \cos \theta = \frac{7}{5} \)**

The cosine function has a range of values between -1 and 1, inclusive. Therefore, any value of cosine must satisfy:
\[ -1 \leq \cos \theta \leq 1 \]
Since \( \frac{7}{5} = 1.4 \) is greater than 1, this expression is not possible.

**Conclusion:** \( \cos \theta = \frac{7}{5} \) is **not possible**.

**Hint:** Check if the value lies within the range of the cosine function, which is between -1 and 1.

2. **Analyzing the second expression: \( \sec \theta = \frac{8}{5} \)**

The secant function is the reciprocal of the cosine function:
\[ \sec \theta = \frac{1}{\cos \theta} \]
The secant function is defined when \( \cos \theta \neq 0 \) and its range is \( (-\infty, -1] \cup [1, \infty) \). Thus, for \( \sec \theta = \frac{8}{5} \), we need:
\[ \cos \theta = \frac{5}{8} \]
Since \( \frac{5}{8} \) is between -1 and 1, this expression is possible.

**Conclusion:** \( \sec \theta = \frac{8}{5} \) is **possible**.

**Hint:** Remember that secant is the reciprocal of cosine and check if the cosine value derived from secant lies within its valid range.

3. **Analyzing the third expression: \( \tan \theta = 45 \)**

The tangent function can take any real value, ranging from \( -\infty \) to \( +\infty \). Therefore, \( \tan \theta = 45 \) is a valid expression.

**Conclusion:** \( \tan \theta = 45 \) is **possible**.

**Hint:** The tangent function has a range of all real numbers, so any finite value is valid.

4. **Final Conclusion:**

After analyzing all the expressions, we find that:
- \( \cos \theta = \frac{7}{5} \) is **not possible**.
- \( \sec \theta = \frac{8}{5} \) is **possible**.
- \( \tan \theta = 45 \) is **possible**.

Therefore, the expressions that are possible are \( \sec \theta = \frac{8}{5} \) and \( \tan \theta = 45 \).

### Summary:
- **Not Possible:** \( \cos \theta = \frac{7}{5} \)
- **Possible:** \( \sec \theta = \frac{8}{5} \) and \( \tan \theta = 45 \)