Question

1. in order to construct a bridge across a lake, an engineer wishes to determine ab. the surveyor found that ac = 530 and m∠c = 43°. if angle a is a right - angle, find the distance across the lake.
2. a scout troop built a frame for a lean - to as shown below. in the right triangle formed by one side of the lean - to, bc = 26 ft and ab = 35 ft. find the m∠b.
3. find the height of the door shown if m∠cab = 65° and ca = 3 ft.
4. points p and q are on the north and south rims, respectively, of the grand canyon. point q is directly south of point p. point r is located 780 m east of p. the m∠prq = 84°. find the width of the canyon.

Explanation:

Step1: 18번 문제 해결

삼각비 $\tan$ 이용. $\tan C=\frac{AB}{AC}$, $AC = 530$, $C = 43^{\circ}$ 이므로 $AB=AC\times\tan C$.
$AB = 530\times\tan43^{\circ}\approx530\times0.9325\approx494.23$ m.

Step2: 19번 문제 해결

삼각비 $\sin$ 이용. $\sin B=\frac{AC}{AB}$, $AC$를 구해야 함. 피타고라스 정리로 $AC=\sqrt{AB^{2}-BC^{2}}=\sqrt{35^{2}-26^{2}}=\sqrt{(35 + 26)(35 - 26)}=\sqrt{61\times9}=\sqrt{549}\approx23.43$. 그러면 $\sin B=\frac{23.43}{35}\approx0.6694$, $B=\sin^{-1}(0.6694)\approx42^{\circ}$.

Step3: 20번 문제 해결

삼각비 $\tan$ 이용. $\tan\angle CAB=\frac{BC}{CA}$, $CA = 3$, $\angle CAB = 65^{\circ}$ 이므로 $BC=CA\times\tan\angle CAB$.
$BC = 3\times\tan65^{\circ}\approx3\times2.1445\approx6.43$ ft.

Step4: 21번 문제 해결

삼각비 $\tan$ 이용. $\tan\angle PRQ=\frac{PQ}{PR}$, $PR = 780$, $\angle PRQ = 84^{\circ}$ 이므로 $PQ=PR\times\tan\angle PRQ$.
$PQ = 780\times\tan84^{\circ}\approx780\times9.5144\approx7421.23$ m.

Answer:

1. $494.23$ m
2. $42^{\circ}$
3. $6.43$ ft
4. $7421.23$ m