Question

A parrel of land has sides measuring175ft, 234ft, 295ft and 415ft and the angle between the sides of the length 234ft and 295ft has measure 137.1° what is the measure of the opposite this angle?

Answer 1

Answer on Question #54463– Math – Trigonometry

Question

A parcel of land has sides measuring 175ft, 234ft, 295ft and 415ft and the angle between the sides of the length 234ft and 295ft has measure $137.1{}^{\circ}$ what is the measure of the opposite this angle?

Solution

According to the statement of the problem, the parcel of land has the quadrangular form (fig. 1).

Fig. 1

We will find the angle $\delta$ , which is opposite to the angle $\beta = 137.1{}^{\circ}$ , in a few steps.

At first we find the length of diagonal $AC$ by using the law of cosines in triangle $ABC$ :

$$(A C) ^ {2} = (A B) ^ {2} + (B C) ^ {2} - 2 A B \cdot B C \cdot \cos \beta .$$

Then using the law of cosines in triangle ADC, we find the cosine of angle $\delta$ :

$$\begin{array}{l} (A C) ^ {2} = (A D) ^ {2} + (D C) ^ {2} - 2 A D \cdot D C \cdot \cos \delta ; \Rightarrow \\ \cos \delta = \frac {(A D) ^ {2} + (D C) ^ {2} - (A C) ^ {2}}{2 A D \cdot D C}. \\ \end{array}$$

Now, substituting (1) into (2) we obtain the general formula

$$\cos \delta = \frac {(A D) ^ {2} + (D C) ^ {2} - (A B) ^ {2} - (B C) ^ {2} + 2 A B \cdot B C \cdot \cos \beta}{2 A D \cdot D C}.$$

The substitution of all measures in (3) gives

$$\cos \delta = \frac {(175)^2 + (415)^2 - (234)^2 - (295)^2 + 2 \cdot 234 \cdot 295 \cdot \cos (137.1{}^\circ)}{2 \cdot 175 \cdot 415} = -0.27584.$$

And finally we use the inverse cosine, to find angle $\delta$:

$$\delta = \arccos(-0.27584) = 106.06815{}^\circ \approx 106.1{}^\circ.$$

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