Question

applying the law of cosines to surveying
a surveyor measures the lengths of the sides of a triangular plot of land. what is the measure of the angle of the triangular plot at which the surveyor stands? approximate to the nearest degree.
$cos^{-1}(0.75)=41^circ$
$cos^{-1}(0.125)=83^circ$
$cos^{-1}(0.563)=56^circ$
$cos^{-1}(0.15)=89^circ$
200 m
250 m
300 m

Explanation:

Step1: Identify Law of Cosines form

For angle $C$ opposite side $c$, $\cos C=\frac{a^2+b^2-c^2}{2ab}$
Here, the surveyor's angle is between 200 m and 250 m, so $a=200$, $b=250$, $c=300$.

Step2: Substitute values into formula

$\cos C=\frac{200^2+250^2-300^2}{2\times200\times250}$

Step3: Calculate numerator and denominator

Numerator: $200^2+250^2-300^2=40000+62500-90000=12500$
Denominator: $2\times200\times250=100000$

Step4: Simplify the fraction

$\cos C=\frac{12500}{100000}=0.125$

Step5: Find inverse cosine

$C=\cos^{-1}(0.125)\approx83^\circ$

Answer:

$\cos^{-1}(0.125) = 83^\circ$