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 $\cot A = \sqrt{5}$, then find the value of $\frac{\csc^2 A - \sec^2 A}{\csc^2 A + \sec^2 A}$
- Find the value of $\frac{\cos^6 \theta + \sin^6 \theta + 3 \sin^2 \theta \cos^2 \theta}{\csc \theta \sec \theta ((\sin \theta + \cos \theta)^2 - 1)}$
- The value of $2\sin30^{\circ} + 5\cos60^{\circ} + 3\sin180^{\circ}$ is:
If $\tan\theta = \frac{5}{12}$, $0 < \theta < \frac{\pi}{2}$, then the value of $\frac{\cos\theta + 5\cot\theta}{\text{cosec}\theta - \cos\theta}$ will be:
- If $\cos^4\theta - \sin^4\theta = \frac{2}{3}$ then the value of $1 - 2\sin^2\theta$ is:
- If $\sin(3A - 20^\circ) = \cos(20^\circ - 3B)$, then the value of A - B is:
- If $\tan A = \frac{8}{15}$ then find $\sin A + \cos A$.
- If $\sin \theta - \cos \theta = 0$, and $0 \le \theta \le 90^\circ$, then the value of $\sin \theta + \cos \theta$ is:
- If $1 + \tan \theta = \sqrt{3}$, then $\sqrt{3} \cot \theta - 1 = ?$
- $\cos(x - y) = \frac{\sqrt{3}}{2}$, and $\sin(x + y) = 1$, where $x$ and $y$ are positive acute angles and $x \ge y$, then $x$ and $y$ are:
Important Questions from Trigonometric Ratios and Identities
If cosec θ = 13/12, then sin θ + cos θ - tan θ is equal to:
What is the value of \(\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}\) ?
If \(\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{k + 1}}{{k - 1}},\) then k = ?
If α + β = 90° and α = 2β, then the value of 3 cos 2 α - 2 sin 2 β is equal to:
If \(\sqrt{3}\) tan θ = 3 sin θ, then what is the value of sin 2θ − cos 2θ ?
