What is the value of \(? = \frac{{ta{n^2}{{60}^0} - 2si{n^2}{{45}^0}}}{{cos{{24}^0}cos{{37}^0}coses{{53}^0}cos{{60}^0}cosec{{66}^0} + si{n^2}{{60}^0}}}\)
Understanding the Trigonometric Expression
The question asks us to find the value of a complex trigonometric expression involving various angles. The expression is given as:
\[ ? = \frac{{ta{n^2}{{60}^0} - 2si{n^2}{{45}^0}}}{{cos{{24}^0}cos{{37}^0}coses{{53}^0}cos{{60}^0}cosec{{66}^0} + si{n^2}{{60}^0}}} \]To solve this, we need to evaluate the numerator and the denominator separately using standard trigonometric values and properties of complementary angles.
Evaluating the Numerator
The numerator is \({\tan^2}{{60}^0} - 2{\sin^2}{{45}^0}\). We need the standard values for \(\tan 60^\circ\) and \(\sin 45^\circ\).
- The value of \(\tan 60^\circ\) is \(\sqrt{3}\).
- The value of \(\sin 45^\circ\) is \(\frac{1}{\sqrt{2}}\).
Now, let's calculate the squared terms:
- \({\tan^2}{{60}^0} = (\sqrt{3})^2 = 3\)
- \({\sin^2}{{45}^0} = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)
Substitute these values into the numerator expression:
\[ \text{Numerator} = 3 - 2 \times \frac{1}{2} = 3 - 1 = 2 \]So, the value of the numerator is 2.
Evaluating the Denominator
The denominator is \({\cos{{24}^0}\cos{{37}^0}\coses{{53}^0}\cos{{60}^0}\cosec{{66}^0} + si{n^2}{{60}^0}}\). This consists of two parts: a product of trigonometric terms and a squared sine term.
Part 1: Product of terms
The first part is \({\cos{{24}^0}\cos{{37}^0}\coses{{53}^0}\cos{{60}^0}\cosec{{66}^0}}\). We can use complementary angle identities here. Recall that:
- \(\cos(90^\circ - \theta) = \sin \theta\)
- \(\cosec(90^\circ - \theta) = \sec \theta\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\cos \theta \sec \theta = 1\)
Observe the angles:
- \(24^\circ + 66^\circ = 90^\circ\)
- \(37^\circ + 53^\circ = 90^\circ\)
We can rewrite \(\cosec 53^\circ\) and \(\cosec 66^\circ\):
- \(\cosec 53^\circ = \cosec(90^\circ - 37^\circ) = \sec 37^\circ\)
- \(\cosec 66^\circ = \cosec(90^\circ - 24^\circ) = \sec 24^\circ\)
Substitute these back into the product:
\[ \cos{{24}^0}\cos{{37}^0}(\sec{{37}^0})\cos{{60}^0}(\sec{{24}^0}) \]Rearrange the terms to group complementary functions:
\[ (\cos{{24}^0}\sec{{24}^0})(\cos{{37}^0}\sec{{37}^0})\cos{{60}^0} \]Using the identity \(\cos \theta \sec \theta = 1\), this simplifies to:
\[ (1)(1)\cos{{60}^0} \]The value of \(\cos 60^\circ\) is \(\frac{1}{2}\).
So, the first part of the denominator is \(1 \times 1 \times \frac{1}{2} = \frac{1}{2}\).
Part 2: Squared sine term
The second part of the denominator is \({\sin^2}{{60}^0}\).
- The value of \(\sin 60^\circ\) is \(\frac{\sqrt{3}}{2}\).
- \({\sin^2}{{60}^0} = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\)
Total Denominator Value
The denominator is the sum of Part 1 and Part 2:
\[ \text{Denominator} = \frac{1}{2} + \frac{3}{4} \]To add these fractions, find a common denominator, which is 4:
\[ \text{Denominator} = \frac{2}{4} + \frac{3}{4} = \frac{2+3}{4} = \frac{5}{4} \]So, the value of the denominator is \(\frac{5}{4}\).
Calculating the Final Value of the Expression
The expression is the numerator divided by the denominator:
\[ ? = \frac{\text{Numerator}}{\text{Denominator}} = \frac{2}{\frac{5}{4}} \]To divide by a fraction, multiply by its reciprocal:
\[ ? = 2 \times \frac{4}{5} = \frac{8}{5} \]The value \(\frac{8}{5}\) can also be written as a mixed number:
\[ \frac{8}{5} = 1 \frac{3}{5} \]Thus, the value of the given trigonometric expression is \(1\frac{3}{5}\).
| Component | Calculation | Value |
|---|---|---|
| Numerator | \({\tan^2}{{60}^0} - 2{\sin^2}{{45}^0} = (\sqrt{3})^2 - 2\left(\frac{1}{\sqrt{2}}\right)^2\) | \(3 - 2(\frac{1}{2}) = 3 - 1 = 2\) |
| Denominator (Part 1) | \({\cos{{24}^0}\cos{{37}^0}\coses{{53}^0}\cos{{60}^0}\cosec{{66}^0}}\) | \((\cos{{24}^0}\sec{{24}^0})(\cos{{37}^0}\sec{{37}^0})\cos{{60}^0} = 1 \times 1 \times \frac{1}{2} = \frac{1}{2}\) |
| Denominator (Part 2) | \({\sin^2}{{60}^0} = \left(\frac{\sqrt{3}}{2}\right)^2\) | \(\frac{3}{4}\) |
| Total Denominator | Part 1 + Part 2 | \(\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}\) |
| Final Value | Numerator / Denominator | \(\frac{2}{5/4} = 2 \times \frac{4}{5} = \frac{8}{5}\) or \(1\frac{3}{5}\) |
Revision Table: Key Trigonometric Values and Identities
| Angle (\(\theta\)) | \(\sin \theta\) | \(\cos \theta\) | \(\tan \theta\) |
|---|---|---|---|
| \(45^\circ\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{\sqrt{2}}\) | 1 |
| \(60^\circ\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) |
| Identity | Description |
|---|---|
| \(\cos(90^\circ - \theta) = \sin \theta\) | Cosine of a complementary angle is sine of the angle. |
| \(\cosec(90^\circ - \theta) = \sec \theta\) | Cosecant of a complementary angle is secant of the angle. |
| \(\sec \theta = \frac{1}{\cos \theta}\) | Secant is the reciprocal of cosine. |
| \(\cosec \theta = \frac{1}{\sin \theta}\) | Cosecant is the reciprocal of sine. |
| \(\cos \theta \sec \theta = 1\) | Product of cosine and secant of the same angle is 1. |
Additional Information on Trigonometry Simplification
Simplifying trigonometric expressions often involves several key strategies:
- Know Standard Angle Values: Be familiar with the sine, cosine, and tangent values for common angles like \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\).
- Use Reciprocal Identities: Replace \(\sec \theta, \cosec \theta, \cot \theta\) with their reciprocal forms (\(\frac{1}{\cos \theta}, \frac{1}{\sin \theta}, \frac{1}{\tan \theta}\)).
- Apply Quotient Identities: Use \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
- Implement Pythagorean Identities: Remember \(\sin^2 \theta + \cos^2 \theta = 1\), \(1 + \tan^2 \theta = \sec^2 \theta\), and \(1 + \cot^2 \theta = \cosec^2 \theta\).
- Recognize Complementary Angles: Angles that sum to \(90^\circ\) (like \(24^\circ\) and \(66^\circ\), or \(37^\circ\) and \(53^\circ\)) have related trigonometric function values (e.g., \(\sin \theta = \cos(90^\circ - \theta)\)).
- Look for Simplification Opportunities: After applying identities, look for terms that cancel out or can be combined.
Practice with different types of expressions helps in identifying which identities are most useful in a given problem.
Important Questions from Trigonometric Functions
If A = cos2θ + sin4θ then for all values of θ is :
The minimum value of 4 cosθ + 3 is
If \(\frac{{\sin x + \cos x}}{{\sin x - \cos x}} = \frac{6}{5}\) , then the value of \(\frac{{{{\tan }^2}x + 1}}{{{{\tan }^2}x - 1}}\) is:
If Y = tan35°, then the value of (2tan55° + cot55°) is :
If -sin θ + cosec θ = 6, then what is the value of sin θ + cosec θ?
