$ \cosec \theta - \sin \theta = p^3 $ and $ \sec \theta - \cos \theta = q^3 $
Solving for tan θ: Given Trigonometric Equations
We are given two trigonometric equations:
- \(\ \ \cosec \theta - \sin \theta = p^3\)
- \(\ \ \sec \theta - \cos \theta = q^3\)
We need to find the value of \(\tan \theta\).
Simplifying the Given Equations
Rewrite the equations using fundamental trigonometric identities (\(\ \cosec \theta = \frac{1}{\sin \theta}\), \(\ \sec \theta = \frac{1}{\cos \theta}\), \(\ 1 - \sin^2 \theta = \cos^2 \theta\), \(\ 1 - \cos^2 \theta = \sin^2 \theta\)).
-
\(\ \frac{1}{\sin \theta} - \sin \theta = p^3\)
\(\implies \frac{1 - \sin^2 \theta}{\sin \theta} = p^3\)
\(\implies \frac{\cos^2 \theta}{\sin \theta} = p^3\) (Equation 1) -
\(\ \frac{1}{\cos \theta} - \cos \theta = q^3\)
\(\implies \frac{1 - \cos^2 \theta}{\cos \theta} = q^3\)
\(\implies \frac{\sin^2 \theta}{\cos \theta} = q^3\) (Equation 2)
Deriving tan θ
To find \(\tan \theta\), we can divide Equation 2 by Equation 1:
\(\frac{q^3}{p^3} = \frac{(\frac{\sin^2 \theta}{\cos \theta})}{(\frac{\cos^2 \theta}{\sin \theta})}\)
Simplify the right side:
\(\frac{q^3}{p^3} = \frac{\sin^2 \theta}{\cos \theta} \times \frac{\sin \theta}{\cos^2 \theta}\)
\(\frac{q^3}{p^3} = \frac{\sin^3 \theta}{\cos^3 \theta}\)
Recognize that \(\ \frac{\sin^3 \theta}{\cos^3 \theta} = (\tan \theta)^3\):
\(\frac{q^3}{p^3} = (\tan \theta)^3\)
Take the cube root of both sides:
\(\tan \theta = \sqrt[3]{\frac{q^3}{p^3}}\)
\(\tan \theta = \frac{q}{p}\)
Conclusion
Therefore, \(\tan \theta\) is equal to \(\frac{q}{p}\). This matches Option B.
Similar Questions
If \(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} = \text{cosec}\theta\), then what is \(\sin\theta\) equal to?
- If \(2 \tan\theta = \sec^2\theta-2\), where \(0 < \theta < \pi /2\), then what is \(\cot\theta\) equal to?
- What is \(\frac{\tan^2 \alpha}{\tan^2 \beta}\) equal to?
- If \(\alpha\) and \(\beta\) are complementary angles, then which one of the following is correct?
- What is \(\sec\theta\) equal to?
- What is \(\tan\theta\) equal to?
- What is the relation between \(p\) and \(q\)?
- What is $\tan\theta + \cot\theta$ equal to?
- Consider the following statements :
1. In a triangle ABC, if \(\sin A + \sin B + \sin C = \frac{3\sqrt{3}}{2}\), then the triangle can be equilateral.
2. In a triangle ABC, if \(\cos A + \cos B + \cos C = \frac{3}{2}\), then the triangle can be equilateral.
Which of the statements given above is/are correct? What is \((1 + \tan\alpha \tan\beta)^2 + (\tan\alpha - \tan\beta)^2\) equal to ?
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