Question

two checkpoints, a and b, in an orienteering course are 5 km apart. another checkpoint, c, is 3 km from checkpoint a. the angle between ab and ac is 25°. determine the distance between checkpoints b and c.

Explanation:

Step1: Identify the law of cosines formula

The law of cosines for finding side $BC$ (let $AB = c = 5$, $AC=b = 3$, $\angle A=\alpha=25^{\circ}$) is $a^{2}=b^{2}+c^{2}-2bc\cos\alpha$.

Step2: Substitute the given values

Substitute $b = 3$, $c = 5$, and $\alpha = 25^{\circ}$ into the formula. First, find $\cos(25^{\circ})\approx0.9063$. Then $a^{2}=3^{2}+5^{2}-2\times3\times5\times0.9063$.
$a^{2}=9 + 25- 27.189$.
$a^{2}=6.811$.

Step3: Solve for $a$

Take the square - root of both sides: $a=\sqrt{6.811}\approx2.61$ km.

Answer:

$2.61$ km