Question

3 formula 2 points given right triangle abc, where side \c\ is the hypotenuse, angle b measures 49 degrees, and side c measures 18, find the length of side b. be sure your calculator is in deg mode, and use the proper trig function in your computation. round your answer to the nearest hundredth. answer 4 formula 2 points a ladder is leaning against a wall so that it makes an angle of 31 degrees with the wall. if the base of the ladder is 6 feet from the bottom of the wall, how long is the ladder?

Explanation:

Question 3 Solution:

Step1: Identify trigonometric ratio

In right triangle \(ABC\), for angle \(B\), \(\sin(B)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{b}{c}\).

Step2: Substitute values

We know \(B = 49^\circ\), \(c = 18\). So \(\sin(49^\circ)=\frac{b}{18}\).

Step3: Solve for \(b\)

Multiply both sides by 18: \(b = 18\times\sin(49^\circ)\).
Using calculator (Deg mode): \(\sin(49^\circ)\approx0.7547\), so \(b\approx18\times0.7547 = 13.5846\approx13.58\).

Step1: Analyze the right triangle

The ladder, wall, and ground form a right triangle. The angle between ladder and wall is \(31^\circ\), the distance from base of ladder to wall (opposite to \(31^\circ\)) is 6 feet, and ladder length is hypotenuse (\(L\)). So \(\sin(31^\circ)=\frac{6}{L}\).

Step2: Solve for \(L\)

Rearrange: \(L=\frac{6}{\sin(31^\circ)}\).
Using calculator (Deg mode): \(\sin(31^\circ)\approx0.5150\), so \(L\approx\frac{6}{0.5150}\approx11.65\).

Answer:

\(13.58\)

Question 4 Solution: