Question

part 1: multiple choice
choose the best answer. show your work in the space to the right for each problem.
self - assess your progress using the check boxes.

1. in the accompanying diagram, what is (sin e)?

2 points
(diagram of triangle (efg) with (ef = 5), (fg = 4), (eg = 3), right - angled at (g))
options:
a. (\frac{3}{5})
b. (\frac{4}{5})
c. (\frac{3}{4})
d. (\frac{4}{3})

1. in the accompanying diagram of right triangle (abc), (angle b) is a right angle, (ab = 8), (bc = 15), and (ca = 17).

2 points
what ratio is equal to (\frac{8}{17})?
(diagram of right - triangle (abc) with right angle at (b), (ab = 8), (bc = 15), (ac = 17))
options:
a. (sin a)
b. (sin c)
c. (cos c)
d. (\tan a)

1. a tree cast a 25 - foot shadow on a sunny day. if the angle of elevation from the tip of the shadow to the top of the tree is (32^{circ}), what is the height of the tree to the nearest tenth of a foot?

2 points
(diagram with a sun, a tree, and a 25 - foot shadow, angle of elevation (32^{circ}))
options:
a. 13.2
b. 15.6
c. 21.2
d. 40.0

Explanation:

Step1: Solve for $\sin E$

For right triangle $EFG$, $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle E$, opposite side is $FG=4$, hypotenuse $FE=5$.
$\sin E = \frac{4}{5}$

Step2: Match $\frac{8}{17}$ to trig ratio

For right triangle $ABC$, $\angle B=90^\circ$. $\sin C=\frac{\text{opposite to } C}{\text{hypotenuse}}=\frac{AB}{CA}=\frac{8}{17}$

Step3: Calculate tree height

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Let $h$ = tree height, $\theta=32^\circ$, adjacent side=25 ft.
$h = 25 \times \tan(32^\circ) \approx 25 \times 0.6249 = 15.6225$
Round to nearest tenth: $15.6$

Answer:

1. d. $\frac{4}{5}$
2. b. $\sin C$
3. b. 15.6