Question

unit review - right triangles and trigonometry
two paths ascend separate hills from the same point. one is 22 ft long, the other 28 ft. a 39 ft bridge connects their endpoints. what is the angle between the paths at the base?
base angle = ?°
round your answer to the nearest hundredth.

Explanation:

Step1: Identify Law of Cosines

We use the Law of Cosines for a triangle with sides $a=22$, $b=28$, $c=39$, solving for angle $C$ opposite side $c$:
$$c^2 = a^2 + b^2 - 2ab\cos(C)$$

Step2: Rearrange to solve for $\cos(C)$

Isolate the cosine term:
$$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$$

Step3: Substitute values into formula

Plug in $a=22$, $b=28$, $c=39$:
$$\cos(C) = \frac{22^2 + 28^2 - 39^2}{2(22)(28)}$$

Step4: Calculate numerator and denominator

Compute each term:
$$22^2=484,\ 28^2=784,\ 39^2=1521$$
$$\text{Numerator} = 484 + 784 - 1521 = -253$$
$$\text{Denominator} = 2(22)(28) = 1232$$
$$\cos(C) = \frac{-253}{1232} \approx -0.20536$$

Step5: Find angle using arccosine

Calculate the inverse cosine:
$$C = \arccos(-0.20536) \approx 101.85^\circ$$

Answer:

$101.85^\circ$