Solving Trigonometric Equation for cotθ
The problem asks us to find the value of \(\cot\theta\) given the equation \(2 \tan\theta = \sec^2\theta - 2\), with the condition that \(0 < \theta < \pi / 2\). This condition means \(\theta\) lies in the first quadrant.
Using Trigonometric Identities
We can use the fundamental trigonometric identity relating \(\sec^2\theta\) and \(\tan^2\theta\): \(\sec^2\theta = 1 + \tan^2\theta\) Substitute this into the given equation:
\(2 \tan\theta = (1 + \tan^2\theta) - 2\)
Simplify the equation:
\(2 \tan\theta = \tan^2\theta - 1\)
Solving the Quadratic Equation
Rearrange the terms to form a quadratic equation in terms of \(\tan\theta\): \(\tan^2\theta - 2 \tan\theta - 1 = 0\) Let \(x = \tan\theta\). The equation becomes: \(x^2 - 2x - 1 = 0\) We can solve this quadratic equation using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a=1\), \(b=-2\), and \(c=-1\).
$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)} $
$x = \frac{2 \pm \sqrt{4 + 4}}{2} $
$x = \frac{2 \pm \sqrt{8}}{2} $
$x = \frac{2 \pm 2\sqrt{2}}{2} $
$x = 1 \pm \sqrt{2} $
So, the possible values for \(\tan\theta\) are \(1 + \sqrt{2}\) and \(1 - \sqrt{2}\).
Determining the Correct Value of tanθ
We are given that \(0 < \theta < \pi / 2\). In this interval (the first quadrant), the value of \(\tan\theta\) is always positive.
- \(1 + \sqrt{2}\) is positive (approximately \(1 + 1.414 = 2.414\)).
- \(1 - \sqrt{2}\) is negative (approximately \(1 - 1.414 = -0.414\)).
Therefore, we must choose the positive value:
\(\tan\theta = 1 + \sqrt{2}\)
Calculating cotθ
The cotangent function is the reciprocal of the tangent function:
\(\cot\theta = \frac{1}{\tan\theta}\)
Substitute the value of \(\tan\theta\) we found:
\(\cot\theta = \frac{1}{1 + \sqrt{2}}\)
To simplify this expression, we rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator (\(1 - \sqrt{2}\)):
$ \cot\theta = \frac{1}{1 + \sqrt{2}} \times \frac{1 - \sqrt{2}}{1 - \sqrt{2}} $
$ \cot\theta = \frac{1 - \sqrt{2}}{1^2 - (\sqrt{2})^2} $
$ \cot\theta = \frac{1 - \sqrt{2}}{1 - 2} $
$ \cot\theta = \frac{1 - \sqrt{2}}{-1} $
$ \cot\theta = -(1 - \sqrt{2}) $
$ \cot\theta = \sqrt{2} - 1 $
Thus, the value of \(\cot\theta\) is \(\sqrt{2} - 1\).
Similar Questions
If \(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} = \text{cosec}\theta\), then what is \(\sin\theta\) 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 ?
Consider the following statements :
I. \(\tan 50^\circ - \cot 50^\circ\) is positive
II. \(\cot 25^\circ - \tan 25^\circ\) is negative
Which of the statements is/are correct?
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 ______
