$ p + q \cot \theta = 3 \cosec \theta $ and $ q - p \cot \theta = 2 \cosec \theta $
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Equations Solution
We are given the two trigonometric equations:
- Equation 1: \(p + q \cot \theta = 3 \cosec \theta\)
- Equation 2: \(q - p \cot \theta = 2 \cosec \theta\)
Trigonometric Transformation
To simplify, we can rewrite the equations using \(\sin \theta\) and \(\cos \theta\). Multiply both sides of each equation by \(\sin \theta\), assuming \(\sin \theta \neq 0\):
Equation 1 multiplied by \(\sin \theta\): \((p + q \frac{\cos \theta}{\sin \theta}) \sin \theta = 3 \frac{1}{\sin \theta} \sin \theta\) \(p \sin \theta + q \cos \theta = 3\) (Eq. 3)
Equation 2 multiplied by \(\sin \theta\): \((q - p \frac{\cos \theta}{\sin \theta}) \sin \theta = 2 \frac{1}{\sin \theta} \sin \theta\) \(q \sin \theta - p \cos \theta = 2\) (Eq. 4)
Squaring and Combining Equations
Now, square both Equation 3 and Equation 4:
Squaring Eq. 3: \((p \sin \theta + q \cos \theta)^2 = 3^2\) \(p^2 \sin^2 \theta + 2pq \sin \theta \cos \theta + q^2 \cos^2 \theta = 9\) (Eq. 5)
Squaring Eq. 4: \((q \sin \theta - p \cos \theta)^2 = 2^2\) \(q^2 \sin^2 \theta - 2pq \sin \theta \cos \theta + p^2 \cos^2 \theta = 4\) (Eq. 6)
Add the results from Eq. 5 and Eq. 6:
\((p^2 \sin^2 \theta + 2pq \sin \theta \cos \theta + q^2 \cos^2 \theta) + (q^2 \sin^2 \theta - 2pq \sin \theta \cos \theta + p^2 \cos^2 \theta) = 9 + 4\)Combine like terms:
\(p^2 \sin^2 \theta + p^2 \cos^2 \theta + q^2 \cos^2 \theta + q^2 \sin^2 \theta = 13\) \(p^2 (\sin^2 \theta + \cos^2 \theta) + q^2 (\cos^2 \theta + \sin^2 \theta) = 13\)Identity \(p^2+q^2\) Result
Apply the fundamental trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\(p^2(1) + q^2(1) = 13\) \(p^2 + q^2 = 13\)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
(secθ + tanθ)/(secθ - tanθ) is equal to:
If tan 45°, cot θ then the value of θ, in radians is
ABC is a triangle If sin (A+B)/2 = √3/2, then the value of sin C/2 is
The angles of elevation of the top of a temple, from the foot and the top of a building 30 m high, are 60° and 30° respectively. Then height of the temple is
The value of sin θ and cos θ when the angle θ is 0 ≤ θ ≤ 90°, is ______
