If `tan theta = (20)/(21),` then `cos theta` is equal to

Options

A. `pm(20)/(41)`
B. `pm(1)/(21)`
C. `pm(21)/(29)`
D. `pm(20)/(21)`

Answer

To find \( \cos \theta \) given that \( \tan \theta = \frac{20}{21} \), we can follow these steps:

### Step 1: Understand the relationship of tangent
We know that:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{20}{21} \]
This means we can represent the opposite side (perpendicular) as 20 and the adjacent side (base) as 21.

### Step 2: Draw a right triangle
We can visualize this as a right triangle where:
- The opposite side (AC) = 20
- The adjacent side (AB) = 21
- The hypotenuse (BC) is unknown.

### Step 3: Use the Pythagorean theorem
According to the Pythagorean theorem:
\[ BC^2 = AC^2 + AB^2 \]
Substituting the values we have:
\[ BC^2 = 20^2 + 21^2 \]
Calculating the squares:
\[ BC^2 = 400 + 441 = 841 \]
Thus, we find:
\[ BC = \sqrt{841} = 29 \]

### Step 4: Calculate \( \cos \theta \)
Now that we have all sides of the triangle, we can find \( \cos \theta \):
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{BC} = \frac{21}{29} \]

### Step 5: Consider the sign of \( \cos \theta \)
Since \( \theta \) is in the first quadrant (where both sine and cosine are positive), we conclude:
\[ \cos \theta = \frac{21}{29} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{21}{29} \]

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