If 2k sin 30° cos 30° cot 60° = \(\frac{{{{\cot }^2}30^\circ \sec 60^\circ \tan 45^\circ }}{{{\rm{cose}}{{\rm{c}}^{\rm{2}}}{\rm{45^\circ cosec 30}}^\circ }}\) , then find the value of k.
3
Solving Trigonometric Equations for 'k'
This problem requires us to find the value of 'k' in a given trigonometric equation. The equation involves trigonometric ratios of standard angles like 30°, 45°, and 60°. To solve this, we need to substitute the known values of these trigonometric ratios and simplify the equation.
Breaking Down the Trigonometric Equation
The given equation is:
\(2k \sin 30^\circ \cos 30^\circ \cot 60^\circ = \frac{{\cot^2 30^\circ \sec 60^\circ \tan 45^\circ}}{{\text{cosec}^2 45^\circ \text{cosec } 30^\circ}}\)
We will evaluate the Left Hand Side (LHS) and the Right Hand Side (RHS) separately by substituting the standard trigonometric values.
Standard Trigonometric Values Needed
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\cot 60^\circ = \frac{1}{\sqrt{3}}\)
- \(\cot 30^\circ = \sqrt{3}\)
- \(\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{1/2} = 2\)
- \(\tan 45^\circ = 1\)
- \(\text{cosec } 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{1/\sqrt{2}} = \sqrt{2}\)
- \(\text{cosec } 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{1/2} = 2\)
Evaluating the Left Hand Side (LHS)
The LHS is \(2k \sin 30^\circ \cos 30^\circ \cot 60^\circ\). Substituting the values:
\(\text{LHS} = 2k \times \left(\frac{1}{2}\right) \times \left(\frac{\sqrt{3}}{2}\right) \times \left(\frac{1}{\sqrt{3}}\right)\)
Simplify the expression:
\(\text{LHS} = k \times \left(\frac{1}{1}\right) \times \left(\frac{\sqrt{3}}{2}\right) \times \left(\frac{1}{\sqrt{3}}\right)\)
\(\text{LHS} = k \times \frac{\sqrt{3}}{2\sqrt{3}}\)
Cancel out \(\sqrt{3}\):
\(\text{LHS} = k \times \frac{1}{2}\)
\(\text{LHS} = \frac{k}{2}\)
Evaluating the Right Hand Side (RHS)
The RHS is \(\frac{{\cot^2 30^\circ \sec 60^\circ \tan 45^\circ}}{{\text{cosec}^2 45^\circ \text{cosec } 30^\circ}}\). Let's evaluate the numerator and denominator separately.
Numerator: \(\cot^2 30^\circ \sec 60^\circ \tan 45^\circ\)
- \(\cot^2 30^\circ = (\sqrt{3})^2 = 3\)
- \(\sec 60^\circ = 2\)
- \(\tan 45^\circ = 1\)
Numerator value = \(3 \times 2 \times 1 = 6\)
Denominator: \(\text{cosec}^2 45^\circ \text{cosec } 30^\circ\)
- \(\text{cosec}^2 45^\circ = (\sqrt{2})^2 = 2\)
- \(\text{cosec } 30^\circ = 2\)
Denominator value = \(2 \times 2 = 4\)
Now, calculate the RHS:
\(\text{RHS} = \frac{\text{Numerator}}{\text{Denominator}} = \frac{6}{4}\)
Simplify the fraction:
\(\text{RHS} = \frac{3}{2}\)
Solving for the Value of k
Now we equate the simplified LHS and RHS:
\(\text{LHS} = \text{RHS}\)
\(\frac{k}{2} = \frac{3}{2}\)
To find 'k', multiply both sides of the equation by 2:
\(k = \frac{3}{2} \times 2\)
\(k = 3\)
Thus, the value of k that satisfies the given trigonometric equation is 3.
Revision Table: Standard Trigonometric Values
| Angle | sin | cos | tan | cosec | sec | cot |
|---|---|---|---|---|---|---|
| 30° | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{3}}\) | 2 | \(\frac{2}{\sqrt{3}}\) | \(\sqrt{3}\) |
| 45° | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{\sqrt{2}}\) | 1 | \(\sqrt{2}\) | \(\sqrt{2}\) | 1 |
| 60° | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) | \(\frac{2}{\sqrt{3}}\) | 2 | \(\frac{1}{\sqrt{3}}\) |
Additional Information: Reciprocal Identities
Understanding reciprocal identities is crucial when working with trigonometric ratios like secant, cosecant, and cotangent. They relate these ratios back to sine, cosine, and tangent:
- \(\text{cosec } \theta = \frac{1}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\)
Using these identities helps simplify complex trigonometric expressions and equations, just like we did in solving for 'k' by converting secant and cosecant values based on cosine and sine values.
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