If cos `67^(@) = tan x^(@)` , then x =

Options

A. `0.4`
B. `6.8`
C. `7.8`
D. 21

Explanation

To solve the equation \( \cos 67^\circ = \tan x^\circ \), we will follow these steps:

### Step 1: Rewrite the equation
Given:
\[ \cos 67^\circ = \tan x^\circ \]
We can rewrite the tangent function in terms of sine and cosine:
\[ \tan x^\circ = \frac{\sin x^\circ}{\cos x^\circ} \]
Thus, we can express our equation as:
\[ \cos 67^\circ = \frac{\sin x^\circ}{\cos x^\circ} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \cos 67^\circ \cdot \cos x^\circ = \sin x^\circ \]

### Step 3: Rearranging the equation
Rearranging the equation, we have:
\[ \sin x^\circ = \cos 67^\circ \cdot \cos x^\circ \]

### Step 4: Use the identity
Using the identity \( \sin x = \cos(90^\circ - x) \), we can express \( \sin x^\circ \) in terms of cosine:
\[ \sin x^\circ = \cos(90^\circ - x^\circ) \]
Thus, we can rewrite our equation as:
\[ \cos(90^\circ - x^\circ) = \cos 67^\circ \cdot \cos x^\circ \]

### Step 5: Use a calculator
To find \( x \), we can directly compute \( \tan^{-1}(\cos 67^\circ) \) using a calculator.

Calculating \( \cos 67^\circ \):
\[ \cos 67^\circ \approx 0.3907 \]
Now, we find \( x \):
\[ x = \tan^{-1}(0.3907) \]

### Step 6: Final calculation
Using a calculator to find \( \tan^{-1}(0.3907) \):
\[ x \approx 21.0^\circ \]

Thus, the value of \( x \) is approximately:
\[ \boxed{21^\circ} \]