Question

math 2201 september 2025 d)solve for θ

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula for finding an angle $\theta$ in a triangle with sides $a = 40.4$, $b = 46.5$, $c = 52.1$ is $\cos\theta=\frac{a^{2}+b^{2}-c^{2}}{2ab}$.
Substitute the values:
\[

$$\begin{align*} a^{2}&=(40.4)^{2}=1632.16\\ b^{2}&=(46.5)^{2}=2162.25\\ c^{2}&=(52.1)^{2}=2714.41\\ 2ab&=2\times40.4\times46.5 = 3777.6 \end{align*}$$

\]
\[

$$\begin{align*} a^{2}+b^{2}-c^{2}&=1632.16 + 2162.25- 2714.41\\ &=3794.41 - 2714.41\\ &=1080 \end{align*}$$

\]
So, $\cos\theta=\frac{1080}{3777.6}\approx0.286$.

Step2: Find the angle $\theta$

$\theta=\cos^{- 1}(0.286)$
Using a calculator, $\theta\approx73.4^{\circ}$

Answer:

$\theta\approx73.4^{\circ}$