Given below are respectively base and hypotenuse of four right angle triangles :
`1` and `sqrt(5),2` and `sqrt(13),3` and `5,4` and `sqrt(41)`
`theta_(1),theta_(2),theta_(3),theta_(4)` are respectively angle included between them . What are the increasing order of these values .
1. `sin theta_(1)`
2. `tan theta_(2)`
3. `cos theta_(3)`
4. `sec theta_(4)`
Choose the correct code among following :

Options

A. `4-1-2-3`
B. `1-4-3-2`
C. `3-1-2-4`
D. `3-1-4-2`

Answer

To solve the problem, we will calculate the values of \( \sin \theta_1 \), \( \tan \theta_2 \), \( \cos \theta_3 \), and \( \sec \theta_4 \) step by step for the given triangles.

### Step 1: Calculate \( \sin \theta_1 \)
- **Given:** Base \( b_1 = 1 \) and Hypotenuse \( h_1 = \sqrt{5} \)
- **Using Pythagorean theorem:**
\[ \text{Perpendicular} (p_1) = \sqrt{h_1^2 - b_1^2} = \sqrt{(\sqrt{5})^2 - 1^2} = \sqrt{5 - 1} = \sqrt{4} = 2 \]
- **Calculate \( \sin \theta_1 \):**
\[ \sin \theta_1 = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{p_1}{h_1} = \frac{2}{\sqrt{5}} \approx 0.894 \]

### Step 2: Calculate \( \tan \theta_2 \)
- **Given:** Base \( b_2 = 2 \) and Hypotenuse \( h_2 = \sqrt{13} \)
- **Using Pythagorean theorem:**
\[ \text{Perpendicular} (p_2) = \sqrt{h_2^2 - b_2^2} = \sqrt{(\sqrt{13})^2 - 2^2} = \sqrt{13 - 4} = \sqrt{9} = 3 \]
- **Calculate \( \tan \theta_2 \):**
\[ \tan \theta_2 = \frac{\text{Perpendicular}}{\text{Base}} = \frac{p_2}{b_2} = \frac{3}{2} = 1.5 \]

### Step 3: Calculate \( \cos \theta_3 \)
- **Given:** Base \( b_3 = 3 \) and Hypotenuse \( h_3 = 5 \)
- **Using Pythagorean theorem:**
\[ \text{Perpendicular} (p_3) = \sqrt{h_3^2 - b_3^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \]
- **Calculate \( \cos \theta_3 \):**
\[ \cos \theta_3 = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{b_3}{h_3} = \frac{3}{5} = 0.6 \]

### Step 4: Calculate \( \sec \theta_4 \)
- **Given:** Base \( b_4 = 4 \) and Hypotenuse \( h_4 = \sqrt{41} \)
- **Using Pythagorean theorem:**
\[ \text{Perpendicular} (p_4) = \sqrt{h_4^2 - b_4^2} = \sqrt{(\sqrt{41})^2 - 4^2} = \sqrt{41 - 16} = \sqrt{25} = 5 \]
- **Calculate \( \sec \theta_4 \):**
\[ \sec \theta_4 = \frac{\text{Hypotenuse}}{\text{Base}} = \frac{h_4}{b_4} = \frac{\sqrt{41}}{4} \approx 1.60075 \]

### Summary of Values
- \( \sin \theta_1 \approx 0.894 \)
- \( \tan \theta_2 = 1.5 \)
- \( \cos \theta_3 = 0.6 \)
- \( \sec \theta_4 \approx 1.60075 \)

### Step 5: Arrange in Increasing Order
Now we can arrange the values in increasing order:
1. \( \cos \theta_3 \approx 0.6 \)
2. \( \sin \theta_1 \approx 0.894 \)
3. \( \tan \theta_2 = 1.5 \)
4. \( \sec \theta_4 \approx 1.60075 \)

Thus, the increasing order is:
\[ \cos \theta_3, \sin \theta_1, \tan \theta_2, \sec \theta_4 \]

### Final Answer
The increasing order of the values is:
\[ 3, 1, 2, 4 \]