Question

3.4: solving for sides
degree mode!!! - \mode\ → highlight \degree\
steps to find a missing side

1. identify the \opposite\, \adjacent\, and \hypotenuse\ sides of your right triangle, based off of the given angle!
2. choose the correct trig function based off of the two given sides (cosine, sine, or tangent!)
3. set up an equation.
4. solve for x!

1.
2.
3.
4.

Explanation:

Step1: Recall tangent formula

For a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Solve for problem 1

In the first right - triangle, $\theta = 32^{\circ}$, the adjacent side to the angle is $17$ and the opposite side is $x$. Using the tangent formula $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, we have $\tan32^{\circ}=\frac{x}{17}$. Then $x = 17\times\tan32^{\circ}$. Since $\tan32^{\circ}\approx0.6249$, $x\approx17\times0.6249 = 10.6233$.

Step3: Solve for problem 2

In the second right - triangle, $\theta = 29^{\circ}$, the adjacent side to the angle is $31$ and the opposite side is $x$. Using the tangent formula $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, we have $\tan29^{\circ}=\frac{x}{31}$. Then $x = 31\times\tan29^{\circ}$. Since $\tan29^{\circ}\approx0.5543$, $x\approx31\times0.5543=17.1833$.

Step4: Solve for problem 3

In the third right - triangle, $\theta = 47^{\circ}$, the hypotenuse is $52$ and we want to find the side opposite the angle $x$. First, we know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. So $\sin47^{\circ}=\frac{x}{52}$, and $x = 52\times\sin47^{\circ}$. Since $\sin47^{\circ}\approx0.7314$, $x\approx52\times0.7314 = 38.0328$.

Step5: Solve for problem 4

In the fourth right - triangle, $\theta = 72^{\circ}$, the adjacent side to the angle is $19$ and the opposite side is $x$. Using the tangent formula $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, we have $\tan72^{\circ}=\frac{x}{19}$. Then $x = 19\times\tan72^{\circ}$. Since $\tan72^{\circ}\approx3.0777$, $x\approx19\times3.0777 = 58.4763$.

Answer:

1. $x\approx10.62$
2. $x\approx17.18$
3. $x\approx38.03$
4. $x\approx58.48$