If `5sec theta-12 cosec theta=0`, find the values of `sec theta, cos theta` and `sin theta`.

Answer

To solve the equation \(5 \sec \theta - 12 \csc \theta = 0\) and find the values of \(\sec \theta\), \(\cos \theta\), and \(\sin \theta\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 5 \sec \theta - 12 \csc \theta = 0 \]
Rearranging gives:
\[ 5 \sec \theta = 12 \csc \theta \]

### Step 2: Substitute the definitions of secant and cosecant
Recall that:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \]
Substituting these into the equation gives:
\[ 5 \cdot \frac{1}{\cos \theta} = 12 \cdot \frac{1}{\sin \theta} \]

### Step 3: Cross-multiply
Cross-multiplying results in:
\[ 5 \sin \theta = 12 \cos \theta \]

### Step 4: Divide both sides by \(\cos \theta\)
Dividing both sides by \(\cos \theta\) (assuming \(\cos \theta \neq 0\)) gives:
\[ \frac{5 \sin \theta}{\cos \theta} = 12 \]
This simplifies to:
\[ \tan \theta = \frac{12}{5} \]

### Step 5: Create a right triangle
Using the tangent value, we can create a right triangle where:
- Opposite side (perpendicular) = 12
- Adjacent side (base) = 5

### Step 6: Calculate the hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Calculating gives:
\[ \text{Hypotenuse}^2 = 12^2 + 5^2 = 144 + 25 = 169 \]
Thus, the hypotenuse is:
\[ \text{Hypotenuse} = \sqrt{169} = 13 \]

### Step 7: Find \(\sec \theta\), \(\cos \theta\), and \(\sin \theta\)
Now we can find the values:
- \(\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{13}{5}\)
- \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{5}{13}\)
- \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{12}{13}\)

### Final Results
Thus, the final values are:
\[ \sec \theta = \frac{13}{5}, \quad \cos \theta = \frac{5}{13}, \quad \sin \theta = \frac{12}{13} \]