Example 9 A vertical mast is supported at the top by two cables reaching from two points, $A$ and $B$. The cable reaching from point $A$ is 43 metres long and is at an angle of $61^{\circ}$ to the horizontal. Point $B$ is 37 metres from the base of the mast. a Find the height of the mast correct to three decimal places. b. Find the angle to the horizontal of the cable reaching from point $B$ to two decimal places.

Step 1 ：We are given a right triangle with the hypotenuse (the cable from point A) and an angle. We can use trigonometry to find the height of the mast. The sine of an angle in a right triangle is defined as the length of the opposite side (the height of the mast in this case) divided by the length of the hypotenuse (the cable from point A). So, we can rearrange the formula to solve for the height of the mast: height = sin(angle) * hypotenuse.

Step 2 ：Given: hypotenuse = 43, angle = 61 degrees

Step 3 ：First, convert the angle from degrees to radians: angle_rad = \(\frac{61 \times \pi}{180}\) = 1.064650843716541 radians

Step 4 ：Then, calculate the height of the mast: height = sin(angle_rad) * hypotenuse = sin(1.064650843716541) * 43 = 37.60864740699402

Step 5 ：Rounding to three decimal places, the height of the mast is approximately 37.609 meters.

Step 6 ：Final Answer: The height of the mast is \(\boxed{37.609}\) meters, correct to three decimal places.