If `tan theta=2`, find the values of other trigonometric ratios.

Answer

To find the values of other trigonometric ratios given that \( \tan \theta = 2 \), we can follow these steps:

### Step 1: Find \( \cot \theta \)
Since \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \), we can express this as:
\[ \tan \theta = 2 = \frac{2}{1} \]
This means that the opposite side is 2 and the adjacent side is 1.

Now, \( \cot \theta \) is the reciprocal of \( \tan \theta \):
\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{2} \]

### Step 2: Use the Pythagorean Identity to find \( \sec \theta \)
We know the identity:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Substituting \( \tan \theta = 2 \):
\[ \sec^2 \theta = 1 + 2^2 = 1 + 4 = 5 \]
Taking the square root:
\[ \sec \theta = \sqrt{5} \]

### Step 3: Find \( \cos \theta \)
Since \( \sec \theta \) is the reciprocal of \( \cos \theta \):
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{\sqrt{5}} \]

### Step 4: Use the Pythagorean Identity to find \( \sin \theta \)
We know the identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos \theta = \frac{1}{\sqrt{5}} \):
\[ \sin^2 \theta + \left(\frac{1}{\sqrt{5}}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{1}{5} = 1 \]
\[ \sin^2 \theta = 1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5} \]
Taking the square root:
\[ \sin \theta = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \]

### Step 5: Find \( \csc \theta \)
Since \( \csc \theta \) is the reciprocal of \( \sin \theta \):
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{5}}{2} \]

### Summary of Values
- \( \tan \theta = 2 \)
- \( \cot \theta = \frac{1}{2} \)
- \( \sec \theta = \sqrt{5} \)
- \( \cos \theta = \frac{1}{\sqrt{5}} \)
- \( \sin \theta = \frac{2}{\sqrt{5}} \)
- \( \csc \theta = \frac{\sqrt{5}}{2} \)