The value of \(\frac{{\rm cose{c^2}30^\circ {{\rm \sin }^2}45^\circ + {{\rm \sec }^2}60^\circ }}{{\rm tan60^\circ \rm cose{c^2}45^\circ - {{\rm \sec }^2}60^\circ \rm tan45^\circ }}\) is:
3/(√3 - 2)
Evaluating the Trigonometric Expression Value
This solution explains how to find the value of the given trigonometric expression involving specific angles.
Key Trigonometric Values Needed
To solve this problem, we first need to recall the values of the trigonometric functions for the angles 30°, 45°, and 60°.
| Trigonometric Function | Angle (\(^\circ\)) | Value | Squared Value (\( ^2 \)) |
|---|---|---|---|
| \(\rm cosec\) | 30 | 2 | 4 |
| \(\rm \sin\) | 45 | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) |
| \(\rm \sec\) | 60 | 2 | 4 |
| \(\rm tan\) | 60 | \(\sqrt{3}\) | 3 |
| \(\rm cosec\) | 45 | \(\sqrt{2}\) | 2 |
| \(\rm tan\) | 45 | 1 | 1 |
Step-by-Step Calculation of the Expression
The expression given is:
\(\frac{{\rm cose{c^2}30^\circ {{\rm \sin }^2}45^\circ + {{\rm \sec }^2}60^\circ }}{{\rm tan60^\circ \rm cose{c^2}45^\circ - {{\rm \sec }^2}60^\circ \rm tan45^\circ }}\)
Numerator Calculation
Let's calculate the value of the numerator first:
\(\rm cose{c^2}30^\circ {{\rm \sin }^2}45^\circ + {{\rm \sec }^2}60^\circ\)
Substitute the known values:
\( = (2)^2 \times \left(\frac{1}{\sqrt{2}}\right)^2 + (2)^2 \)
Simplify the squares:
\( = 4 \times \frac{1}{2} + 4 \)
Perform the multiplication:
\( = 2 + 4 \)
Result of the numerator:
\( = 6 \)
Denominator Calculation
Now, let's calculate the value of the denominator:
\(\rm tan60^\circ \rm cose{c^2}45^\circ - {{\rm \sec }^2}60^\circ \rm tan45^\circ\)
Substitute the known values:
\( = (\sqrt{3}) \times (\sqrt{2})^2 - (2)^2 \times (1) \)
Simplify the squares and multiplication:
\( = \sqrt{3} \times 2 - 4 \times 1 \)
\( = 2\sqrt{3} - 4 \)
Calculating the Final Expression Value
Now, divide the calculated numerator by the calculated denominator:
\(\text{Expression Value} = \frac{\text{Numerator}}{\text{Denominator}} = \frac{6}{2\sqrt{3} - 4}\)
We can simplify this fraction by factoring out a 2 from the denominator:
\( = \frac{6}{2(\sqrt{3} - 2)} \)
Cancel out the common factor of 2:
\( = \frac{3}{\sqrt{3} - 2} \)
Comparing the Result with Options
The calculated value of the expression is \(\frac{3}{\sqrt{3} - 2}\).
Comparing this result with the given options:
- Option 1: \(\frac{-3}{2 + \sqrt{3}}\)
- Option 2: \(\frac{3}{\sqrt{3} - 2}\)
- Option 3: \(\frac{3}{2 + \sqrt{3}}\)
- Option 4: \(\frac{-3}{\sqrt{3} - 2}\)
Our calculated value \(\frac{3}{\sqrt{3} - 2}\) directly matches Option 2.
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