Solving Sec θ - Cosec θ = 0
We are given the equation $\sec \theta - \text{cosec} \theta = 0$.
- Rewrite the equation as $\sec \theta = \text{cosec} \theta$.
- Express in terms of $\sin \theta$ and $\cos \theta$: $\frac{1}{\cos \theta} = \frac{1}{\sin \theta}$.
- This implies $\sin \theta = \cos \theta$.
- Dividing both sides by $\cos \theta$ (assuming $\cos \theta \neq 0$), we get $\frac{\sin \theta}{\cos \theta} = 1$.
- Therefore, $\tan \theta = 1$.
- The principal value for $\theta$ that satisfies $\tan \theta = 1$ is $\theta = 45^\circ$ or $\frac{\pi}{4}$ radians.
Evaluating the Expression
We need to find the value of $2\sin 2\theta - 3\cos 2\theta$. Substitute $\theta = 45^\circ$ into the expression.
- First, calculate $2\theta$: $2\theta = 2 \times 45^\circ = 90^\circ$.
- Now calculate $\sin 2\theta$ and $\cos 2\theta$:
- $\sin 2\theta = \sin 90^\circ = 1$
- $\cos 2\theta = \cos 90^\circ = 0$
- Substitute these values back into the expression: $2\sin 2\theta - 3\cos 2\theta = 2(1) - 3(0)$
- Simplify the expression: $2 - 0 = 2$.
The value of the expression is $2$.
Similar Questions
- If $\cos(x + y) = \frac{1}{2}$ and $\sin(x - y) = 0$, where x and y are positive acute angles and $x \ge y$, then x and y are:
- If $P = \tan\theta + \sec\theta$, then the value of $\tan\theta$ is:
- If $\tan A = \frac{3}{4}$, then the value of $\frac{\cos A - \sin A}{\cos A + \sin A}$ is:
- What is the value of $\sin(48^\circ + \theta) - \cos(42^\circ - \theta)$?
- What is the value of the following expression?
$(\tan 2^\circ \tan 88^\circ) (\tan 3^\circ \tan 87^\circ) ... (\tan 43^\circ \tan 47^\circ) \tan 45^\circ$ - If the ratio of the sine of an acute angle to its cosine is 5 : 12, then what will the value of sine of that angle be?
- If $\tan\theta + \cot\theta = 6$, then find the value of $\tan^2\theta + \cot^2\theta$
- What is the value of the following expression?
$\frac{\cos 3x + \cos x}{\sin 3x - \sin x}$ - If $\sin(A - B) = \frac{1}{2}$ and $\cos(A + B) = \frac{1}{2}$ with $0^\circ < (A + B) \leq 90^\circ$, $A > B$, then find the measure of A and B.
- If $\sqrt{2} \sin(5x - 5)^\circ = \tan 45^\circ$, then the value of x (in degrees) is:
Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
