Let's solve the problems shown in the image step by step.

Problem 14:

• Triangle $\triangle ABC$, where:
  • $\angle CAB = 47^\circ$,
  • $AB = 3$ (hypotenuse),
  • Find $AC = x$ (adjacent side).

Solution:

Using the cosine function: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ $\cos(47^\circ) = \frac{x}{3}$ $x = 3 \cdot \cos(47^\circ)$ Using $\cos(47^\circ) \approx 0.682$: $x \approx 3 \cdot 0.682 = 2.046$ $\boxed{x \approx 2.05}$

Problem 17:

• Triangle with sides:
  • Adjacent side = 4,
  • Opposite side = 8,
  • Find the hypotenuse $x$.

Solution:

Using the Pythagorean theorem: $x^2 = 4^2 + 8^2$ $x^2 = 16 + 64 = 80$ $x = \sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5} \approx 8.944$ $\boxed{x \approx 8.94}$

Problem 20:

• Triangle $\triangle ABC$, where:
  • $AC = 7.7$,
  • $AB = 14$,
  • Find $\theta$.

Solution:

Using the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $\sin(\theta) = \frac{7.7}{14}$ $\sin(\theta) \approx 0.55$ Using the inverse sine: $\theta = \arcsin(0.55)$ $\theta \approx 33.4^\circ$ $\boxed{\theta \approx 33.4^\circ}$

Let me know if you'd like more detailed explanations! 😊

Follow-up Questions:

1. What other trigonometric functions can be used to solve Problem 14?
2. How would you solve Problem 17 if given a missing side instead of the hypotenuse?
3. What is the tangent value of $\theta$ in Problem 20?
4. How can the angle in Problem 20 be verified with another trigonometric ratio?
5. What is the area of the triangle in Problem 17?

Tip: Remember to keep track of your calculator settings (degrees vs. radians) when solving trigonometric problems!