Let's solve the right triangle based on the given information.

Given dataLet's solve the right triangle based on the given information.

Given

• Angle $A = 35^\circ 32'$
• Angle $B = 54^\circ 28'$ (since it’s a right triangle, the sum of angles $A$ and $B$ should be $90^\circ$).
• Side $c = 966 \, \text{m}$ (the hypotenuse).

We will find sides $a$ (opposite to angle $A$) and $b$ (opposite to angle $B$) using trigonometry.

Step 1: Convert angles to decimal form

• $A = 35^\circ 32'$

  • $32'$ is $\frac{32}{60} = 0.5333^\circ$, so $A = 35.5333^\circ$.

• $B = 54^\circ 28'$

  • $28'$ is $\frac{28}{60} = 0.4667^\circ$, so $B = 54.4667^\circ$.

Step 2: Use sine and cosine functions

1. For side $a$ (opposite to angle $A$):

$\sin(A) = \frac{a}{c}$ $a = c \cdot \sin(A) = 966 \cdot \sin(35.5333^\circ)$

2. For side $b$ (opposite to angle $B$):

$\sin(B) = \frac{b}{c}$ $b = c \cdot \sin(B) = 966 \cdot \sin(54.4667^\circ)$

Now, let's calculate the values of $a$ and $b$.The values for the sides of the triangle are:

• $a = 561 \, \text{m}$
• $b = 786 \, \text{m}$

If you have further questions or need additional details, feel free to ask!

Here are some related questions to deepen understanding:

1. How do you calculate angles using inverse trigonometric functions?
2. What is the difference between degrees and radians in trigonometry?
3. How do sine and cosine relate to the sides of a right triangle?
4. What is the significance of rounding in trigonometric calculations?
5. How can the Pythagorean theorem verify the results of trigonometric calculations?

Tip: When solving triangles, always check if the angles add up to 180° as a quick verification step.