Let 12 $ (\tan \theta + \cot \theta) = 25 $, where $ 45^\circ < \theta < 90^\circ $
Simplify tanθ and cotθ Equation
Given the equation \(12 (\tan \theta + \cot \theta) = 25\) and the range \(45^\circ < \theta < 90^\circ\). We need to find the value of \((\sin \theta - \cos \theta)\).
Rewrite \(\tan \theta\) and \(\cot \theta\) using \(\sin \theta\) and \(\cos \theta\):
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
Substitute into the equation:
\(12 \left( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \right) = 25\)
Find a common denominator:
\(12 \left( \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \right) = 25\)
Apply the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\(12 \left( \frac{1}{\sin \theta \cos \theta} \right) = 25\)
\(\frac{12}{\sin \theta \cos \theta} = 25\)
Solve for the product \(\sin \theta \cos \theta\):
\(\sin \theta \cos \theta = \frac{12}{25}\)
Calculate (sinθ - cosθ)² Value
Consider the square of the target expression \((\sin \theta - \cos \theta)\):
\((\sin \theta - \cos \theta)^2 = \sin^2 \theta - 2 \sin \theta \cos \theta + \cos^2 \theta\)
Group terms and use the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\((\sin \theta - \cos \theta)^2 = (\sin^2 \theta + \cos^2 \theta) - 2 \sin \theta \cos \theta\)
\((\sin \theta - \cos \theta)^2 = 1 - 2 \sin \theta \cos \theta\)
Substitute the value of \(\sin \theta \cos \theta\):
\((\sin \theta - \cos \theta)^2 = 1 - 2 \left( \frac{12}{25} \right) = 1 - \frac{24}{25}\)
\((\sin \theta - \cos \theta)^2 = \frac{25 - 24}{25} = \frac{1}{25}\)
Determine (sinθ - cosθ) Sign
Take the square root of both sides:
\(\sin \theta - \cos \theta = \pm \sqrt{\frac{1}{25}} = \pm \frac{1}{5}\)
Analyze the sign based on the given range \(45^\circ < \theta < 90^\circ\):
- In this range, \(\theta\) is in the first quadrant.
- \(\sin \theta\) and \(\cos \theta\) are both positive.
- For angles greater than \(45^\circ\) (up to \(90^\circ\)), \(\sin \theta > \cos \theta\).
Therefore, \((\sin \theta - \cos \theta)\) must be positive.
Final (sinθ - cosθ) Result
Select the positive value:
\(\sin \theta - \cos \theta = \frac{1}{5}\)
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 ?
Important Questions from Trigonometry
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