Let's break down the question step-by-step to fully understand and solve the given expression:
We need to evaluate the following trigonometric expression:
[tex]\[ \tan^2 30^\circ + 2 \sin 60^\circ + \tan 45^\circ - \tan 60^\circ + \cos^2 30^\circ \][/tex]
### Step 1: Evaluate each trigonometric term individually
1. Calculate [tex]\(\tan^2 30^\circ\)[/tex]:
[tex]\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \][/tex]
[tex]\[ \tan^2 30^\circ = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3} \approx 0.3333 \][/tex]
2. Calculate [tex]\(\sin 60^\circ\)[/tex]:
[tex]\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.8660 \][/tex]
3. Calculate [tex]\(\tan 45^\circ\)[/tex]:
[tex]\[ \tan 45^\circ = 1 \approx 1 \][/tex]
4. Calculate [tex]\(\tan 60^\circ\)[/tex]:
[tex]\[ \tan 60^\circ = \sqrt{3} \approx 1.7321 \][/tex]
5. Calculate [tex]\(\cos^2 30^\circ\)[/tex]:
[tex]\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \][/tex]
[tex]\[ \cos^2 30^\circ = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \approx 0.7500 \][/tex]
### Step 2: Substitute all the values into the expression
Take all the calculated values and substitute them back into the original expression:
[tex]\[ \frac{1}{3} + 2 \left( \frac{\sqrt{3}}{2} \right) + 1 - \sqrt{3} + \frac{3}{4} \][/tex]
Now substitute the approximate numerical values:
[tex]\[ 0.3333 + 2 \times 0.8660 + 1 - 1.7321 + 0.7500 \][/tex]
### Step 3: Simplify the expression
Let's add all these values together step-by-step:
1. [tex]\(0.3333 + 2 \times 0.8660 = 0.3333 + 1.7320 = 2.0653\)[/tex]
2. [tex]\(2.0653 + 1 = 3.0653\)[/tex]
3. [tex]\(3.0653 - 1.7321 = 1.3332\)[/tex]
4. [tex]\(1.3332 + 0.7500 = 2.0832\)[/tex]
### Final Answer
So, the value of the given trigonometric expression is approximately:
[tex]\[ 2.0833 \][/tex]
Given that:
[tex]\[ \frac{25}{12} \approx 2.0833 \][/tex]
Thus:
[tex]\[ \tan^2 30^\circ + 2 \sin 60^\circ + \tan 45^\circ - \tan 60^\circ + \cos^2 30^\circ = \frac{25}{12} \][/tex]
This confirms the answer to the problem.
