Question

find the indicated trigonometric ratio.

1. tanθ
2. sinθ
3. cosθ
4. tanθ
5. sinθ
6. cosθ

Explanation:

Step1: Recall trigonometric ratio formula

For a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

1. For \(\tan\theta\) in the first triangle:

Step1: Identify sides

Opposite side to \(\theta\) is \(20\), adjacent side to \(\theta\) is \(21\)

Step2: Apply formula

\(\tan\theta=\frac{20}{21}\)

2. For \(\sin\theta\) in the second triangle:

Step1: Identify sides

Opposite side to \(\theta\) is \(8\), hypotenuse: \(17\) (using Pythagoras \(15^{2}+8^{2}=225 + 64=289=17^{2}\))

Step2: Apply formula

\(\sin\theta=\frac{8}{17}\)

3. For \(\cos\theta\) in the third triangle:

Step1: Identify sides

Adjacent side to \(\theta\) is \(12\), hypotenuse is \(15\) (using Pythagoras \(12^{2}+9^{2}=144 + 81=225=15^{2}\))

Step2: Apply formula

\(\cos\theta=\frac{12}{15}=\frac{4}{5}\)

4. For \(\tan\theta\) in the fourth triangle:

Step1: Identify sides

Opposite side to \(\theta\) is \(\sqrt{2}\), adjacent side to \(\theta\) is \(\sqrt{8}\)

Step2: Simplify

\(\tan\theta=\frac{\sqrt{2}}{\sqrt{8}}=\frac{\sqrt{2}}{2\sqrt{2}}=\frac{1}{2}\)

5. For \(\sin\theta\) in the fifth triangle:

Step1: Identify sides

Opposite side to \(\theta\) is \(4\), hypotenuse is \(\sqrt{17}\)

Step2: Apply formula

\(\sin\theta=\frac{4}{\sqrt{17}}=\frac{4\sqrt{17}}{17}\)

6. For \(\cos\theta\) in the sixth triangle:

Step1: Identify sides

Adjacent side to \(\theta\) is \(16\), hypotenuse is \(20\)

Step2: Apply formula

\(\cos\theta=\frac{16}{20}=\frac{4}{5}\)

Answer:

1. \(\frac{20}{21}\)
2. \(\frac{8}{17}\)
3. \(\frac{4}{5}\)
4. \(\frac{1}{2}\)
5. \(\frac{4\sqrt{17}}{17}\)
6. \(\frac{4}{5}\)