Sure, let's evaluate the given expression step-by-step:
(a) The expression to evaluate is:
[tex]\[3 \cos^2 30^\circ + \sec^2 30^\circ + 2 \cos^2 45^\circ - 3 \sin^2 30^\circ - \tan^2 60^\circ.\][/tex]
First, we should recall the values of the trigonometric functions involved:
- [tex]\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\)[/tex]
- [tex]\(\cos 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin 30^\circ = \frac{1}{2}\)[/tex]
- [tex]\(\tan 60^\circ = \sqrt{3}\)[/tex]
Now, let’s proceed to calculate the individual terms:
1. [tex]\(3 \cos^2 30^\circ\)[/tex]:
[tex]\[
3 \cos^2 30^\circ = 3 \left( \frac{\sqrt{3}}{2} \right)^2 = 3 \cdot \frac{3}{4} = \frac{9}{4} \approx 2.25
\][/tex]
2. [tex]\(\sec^2 30^\circ\)[/tex]:
[tex]\[
\sec^2 30^\circ = \left( \frac{2\sqrt{3}}{3} \right)^2 = \frac{4 \cdot 3}{9} = \frac{12}{9} = \frac{4}{3} \approx 1.33
\][/tex]
3. [tex]\(2 \cos^2 45^\circ\)[/tex]:
[tex]\[
2 \cos^2 45^\circ = 2 \left( \frac{\sqrt{2}}{2} \right)^2 = 2 \cdot \frac{2}{4} = 2 \cdot \frac{1}{2} = 1
\][/tex]
4. [tex]\(-3 \sin^2 30^\circ\)[/tex]:
[tex]\[
-3 \sin^2 30^\circ = -3 \left( \frac{1}{2} \right)^2 = -3 \cdot \frac{1}{4} = -\frac{3}{4} \approx -0.75
\][/tex]
5. [tex]\(-\tan^2 60^\circ\)[/tex]:
[tex]\[
-\tan^2 60^\circ = -(\sqrt{3})^2 = -3
\][/tex]
Now, let's sum these evaluated terms:
[tex]\[
\begin{align*}
3 \cos^2 30^\circ + \sec^2 30^\circ + 2 \cos^2 45^\circ - 3 \sin^2 30^\circ - \tan^2 60^\circ & = 2.25 + 1.33 + 1 - 0.75 - 3 \\
& = 2.25 + 1.33 + 1 - 0.75 - 3 \\
& \approx 2.25 + 1.33 + 1 - 0.75 - 3 \\
& \approx 0.83
\end{align*}
\][/tex]
Therefore, the evaluated value of the expression is approximately [tex]\(0.83\)[/tex].
