Answer to Question #225050 in Trigonometry for tanmay000

Question

1. Two lighthouses A and B are 25km apart and A is due west of B. A submarine S is on a bearing of 137degrees from A and on a bearing of 170degrees from B. Find the distance of S from A and the distance of S from B.

2. In the triangle PQR, PQ= 7cm, PR= 8cm and QR= 9cm. Find angle QPR.\

3. Find the smallest angle in a triangle whose sides are of length 3x, 4x and 6x.

4. Shop Y is 9km due North of shop Z. Shop X is 8km from Y, 5km from Z and somewhere to the west of line YZ.

a) Draw the triangle XYZ and find angle YZX.

b) To improve sales, shop X is moved due to South until it is due West of Z. Find how far it moved.

Answer 1

1.

Using Sine rule:

$\frac{a}{sin\left(A\right)}=\frac{b}{sin\left(B\right)}$

$\frac{25}{sin\:33}=\frac{b}{sin\:47}$

$b=\frac{25\times sin\:47}{sin\:33}$

$b=33.57km$

Hence, distance of S to B is 33.57km.

$\frac{b}{sin\left(B\right)}=\frac{c}{sin\left(C\right)}$

$\frac{33.57}{sin\left(47\right)}=\frac{c}{sin\left(100\right)}$

$c=\frac{33.57\times sin\:100}{sin\:47}$

$c=45.20km$

Hence, the distance of S to A is 45.20km.

2.

To find angle QPR we use the cosine rule:

$a^2\:=b^2+c^2-2bcCosA$

$Cos\:A=\frac{b^2+c^2-a^2}{2bc}$

$Cos\:A=\frac{7^2+8^2-9^2}{2\times 7\times 8}$

$Cos\:A=0.2857$

$A=Cos^{-1}\left(0.2857\right)$

$A\approx 73.4^o$

Hence angle QPR is $73.4^o$

3.

$Angle\:1=3x$

$Angle\:2=4x$

$Angle\:3=6x$

$Angle\:sum\:property\:of\:a\:triangle=180$

$\:\:3x+6x+4x=180$

$13x=180$

$x=13.85$

The smallest angle will be: $3x$

$3x=3\times13.85\approx41.5^o$

4.

(a). The angle YZX can be determined by cosine rule:

$a^2\:=b^2+c^2-2bcCosA$

$Cos\:A=\frac{b^2+c^2-a^2}{2bc}$

$Cos\:A=\frac{5^2+9^2-8^2}{2\times 5\times 9}$

$A=Cos^{-1}\left(0.4667\right)$

$A\approx62.2^o$

Hence, angle YZX is $62.2^o$

(b).

To determine the distance Shop X moved to a reach a new location we use the formula:

$Distance=\frac{\theta }{360}\:\times 2\times \pi \times r\:$

$Distance=\frac{207.8^o}{360^o}\:\times \:2\times \:\frac{22}{7}\:\times \:5$

$Distance\approx 18.14km$

Hence, shop X moved 18.14 km to reach a new location.