Question

1. \\( \cot \theta \\) 8) \\( \tan \theta \\) 9) \\( \tan \theta \\) 10) \\( \cot \theta \\) 11) \\( \tan \theta \\) 12) \\( \cot \theta \\)

Explanation:

Step1: Recall the definition of cotangent

In a right - triangle, \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\)

For problem 7:
The adjacent side to \(\theta\) is \(20\) and the opposite side is \(15\)
\(\cot\theta=\frac{20}{15}=\frac{4}{3}\)

Step2: Recall the definition of tangent

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

For problem 8:
The opposite side to \(\theta\) is \(22\) and the adjacent side is \(2\sqrt{23}\)
\(\tan\theta=\frac{22}{2\sqrt{23}}=\frac{11}{\sqrt{23}}=\frac{11\sqrt{23}}{23}\)

Step3: For problem 9:

The opposite side to \(\theta\) is \(6\) and the adjacent side is \(8\)
\(\tan\theta=\frac{6}{8}=\frac{3}{4}\)

Step4: For problem 10:

The adjacent side to \(\theta\) is \(2\sqrt{5}\) and the opposite side is \(4\)
\(\cot\theta=\frac{2\sqrt{5}}{4}=\frac{\sqrt{5}}{2}\)

Step5: For problem 11:

The opposite side to \(\theta\) is \(3\) and the adjacent side is \(4\)
\(\tan\theta=\frac{3}{4}\)

Step6: For problem 12:

The adjacent side to \(\theta\) is \(12\) and the opposite side is \(24\)
\(\cot\theta=\frac{12}{24}=\frac{1}{2}\)

Answer:

1. \(\frac{4}{3}\)
2. \(\frac{11\sqrt{23}}{23}\)
3. \(\frac{3}{4}\)
4. \(\frac{\sqrt{5}}{2}\)
5. \(\frac{3}{4}\)
6. \(\frac{1}{2}\)