Question

algebra 2
period: ____ date: ____

1. which equation could be used to find the measure of one acute angle in the right triangle at the right?

a) $\tan\theta=\frac{18}{9}$
b) $cos\theta=\frac{9}{18}$
c) $\tan\theta=\frac{9}{18}$
d) $sin\theta=\frac{9}{18}$

1. in $\triangle def$ is a right triangle with $\angle f = 90^\circ$ and $\tan e=\frac{7}{24}$, which of the following represents $\sec d$?

a) $\frac{25}{24}$
b) $\frac{25}{7}$
c) $\frac{24}{7}$
d) $\frac{7}{24}$

1. a hot air balloon is attached to a 250-ft rope. the rope makes a 58° angle with the ground. approximately how high is the balloon above the ground?

a) 132ft
b) 205ft
c) 212ft
d) 250ft

1. which of the following is $\cos x$ written as a fraction in lowest terms?

a) $\frac{12}{13}$
b) $\frac{5}{12}$
c) $\frac{5}{13}$
d) $\frac{12}{5}$

Explanation:

Step1: Identify trig ratios for θ

For angle $\theta$, opposite side $=9$, adjacent side $=18$, hypotenuse $=\sqrt{9^2+18^2}$ (not needed here). $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{9}{18}$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Match to option 1

Only option c matches $\tan\theta=\frac{9}{18}$.

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Step1: Relate tan E to triangle sides

$\tan E=\frac{7}{24}=\frac{\text{opposite}}{\text{adjacent}}=\frac{DF}{FE}$. Let $DF=7k$, $FE=24k$.

Step2: Find hypotenuse DE

By Pythagoras: $DE=\sqrt{(7k)^2+(24k)^2}=25k$.

Step3: Calculate sec D

$\sec D=\frac{\text{hypotenuse}}{\text{adjacent to }D}=\frac{DE}{DF}=\frac{25k}{7k}=\frac{25}{7}$.

Step4: Match to option 2

This matches option b.

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Step1: Define trig relation for height

Height $h$ is opposite the $58^\circ$ angle, rope length (250 ft) is hypotenuse. Use $\sin\theta=\frac{h}{\text{hypotenuse}}$.

Step2: Solve for h

$h=250\times\sin(58^\circ)$. $\sin(58^\circ)\approx0.8480$, so $h\approx250\times0.8480=212$.

Step3: Match to option 3

This matches option c.

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Step1: Identify sides for cos x

For angle $x$, adjacent side $=24$, hypotenuse $=26$. $\cos x=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{26}$.

Step2: Simplify the fraction

$\frac{24}{26}=\frac{12}{13}$.

Step3: Match to option 4

This matches option a.

Answer:

1. c) $\tan\theta = \frac{9}{18}$
2. b) $\frac{25}{7}$
3. c) 212ft
4. a) $\frac{12}{13}$