Question

1. use inverse trig to find the missing angles.
2. complete the trig ratios using the words opposite, adjacent, and hypotenuse

sin(θ) = —————— cos(θ) = ———————— tan(θ) = ————————

1. finish my work correctly. use regular trig or inverse trig when necessary.

a. solve the problems below, rounding to the nearest degree.
sin(θ) = 0.7891 cos(θ) = 0.1337
b. solve the problems below, rounding to the nearest tenth.
sin(27) = 15/x tan(42) = x/21

Explanation:

4a.

Step1: Identify the trigonometric ratio

We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the opposite side to the unknown angle is \(16\) and the hypotenuse is \(34\). So, \(\sin\theta=\frac{16}{34}\approx0.4706\)

Step2: Use inverse - sine function

\(\theta=\sin^{- 1}(0.4706)\)
Using a calculator, \(\theta\approx28^{\circ}\)

4b.

Step1: Identify the trigonometric ratio

We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, the adjacent side to the unknown angle is \(4\) and the hypotenuse is \(5\). So, \(\cos\theta=\frac{4}{5} = 0.8\)

Step2: Use inverse - cosine function

\(\theta=\cos^{-1}(0.8)\)
Using a calculator, \(\theta\approx37^{\circ}\)

5.

By the definitions of trigonometric ratios in a right - triangle:

• \(\sin(\theta)=\frac{\text{Opposite}}{\text{Hypotenuse}}\)
• \(\cos(\theta)=\frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• \(\tan(\theta)=\frac{\text{Opposite}}{\text{Adjacent}}\)

6a.

Step1: Solve \(\sin(\theta)=0.7891\)

Using the inverse - sine function \(\theta=\sin^{-1}(0.7891)\)
Using a calculator, \(\theta\approx52^{\circ}\)

Step2: Solve \(\cos(\theta)=0.1337\)

Using the inverse - cosine function \(\theta=\cos^{-1}(0.1337)\)
Using a calculator, \(\theta\approx82^{\circ}\)

6b.

Step1: Solve \(\sin(27)=\frac{15}{x}\)

Cross - multiply: \(x\sin(27)=15\)
Then \(x=\frac{15}{\sin(27)}\)
Since \(\sin(27)\approx0.4540\), \(x=\frac{15}{0.4540}\approx33.0\)

Step2: Solve \(\tan(42)=\frac{x}{21}\)

Cross - multiply: \(x = 21\tan(42)\)
Since \(\tan(42)\approx0.9004\), \(x=21\times0.9004\approx18.9\)

Answer:

4a. The missing angle is approximately \(28^{\circ}\)
4b. The missing angle is approximately \(37^{\circ}\)

1. \(\sin(\theta)=\frac{\text{Opposite}}{\text{Hypotenuse}}\), \(\cos(\theta)=\frac{\text{Adjacent}}{\text{Hypotenuse}}\), \(\tan(\theta)=\frac{\text{Opposite}}{\text{Adjacent}}\)

6a. For \(\sin(\theta)=0.7891\), \(\theta\approx52^{\circ}\); for \(\cos(\theta)=0.1337\), \(\theta\approx82^{\circ}\)
6b. For \(\sin(27)=\frac{15}{x}\), \(x\approx33.0\); for \(\tan(42)=\frac{x}{21}\), \(x\approx18.9\)