This is a right triangle problem where you are solving for the lengths of the sides $a$ and $b$, given the angle $A = 54.1^\circ$ and the hypotenuse $c = 55$. Let’s break this down step by step.

Step 1: Identify trigonometric relationships

• From the triangle:
  • $a$ (opposite side) is related to $\sin(A)$:
    $\sin(A) = \frac{a}{c}$
  • $b$ (adjacent side) is related to $\cos(A)$:
    $\cos(A) = \frac{b}{c}$

Step 2: Solve for $a$ and $b$

1. Solve for $a$: $a = c \cdot \sin(A)$ Substituting $c = 55$ and $A = 54.1^\circ$: $a = 55 \cdot \sin(54.1^\circ)$ Using $\sin(54.1^\circ) \approx 0.809$: $a \approx 55 \cdot 0.809 = 44.495$ Rounding to the nearest hundredth: $a \approx 44.50$

2. Solve for $b$: $b = c \cdot \cos(A)$ Substituting $c = 55$ and $A = 54.1^\circ$: $b = 55 \cdot \cos(54.1^\circ)$ Using $\cos(54.1^\circ) \approx 0.588$: $b \approx 55 \cdot 0.588 = 32.34$ Rounding to the nearest hundredth: $b \approx 32.34$

Final Answer:

• $a \approx 44.50$
• $b \approx 32.34$

Would you like a detailed explanation of these calculations? Let me know if you have questions!

Related Questions:

1. What is the general formula for solving sides of a right triangle given one angle and the hypotenuse?
2. How do you verify results in a right triangle using the Pythagorean theorem?
3. What is the significance of rounding to the nearest hundredth in trigonometric calculations?
4. Can we solve for the missing angle in this triangle? (Hint: Sum of angles in a triangle is 180°.)
5. How can we use a scientific calculator to find $\sin$ and $\cos$ values?

Tip:

When solving trigonometric problems, always ensure your calculator is in the correct mode (degrees or radians) depending on the angle given!