The number of solutions of cos4θ – 2cos2θ + 3sin6θ + 1 = 0 in [0,2π] is
1
2
3
4
The Correct Option is C
Solution and Explanation
The correct answer is option (C): 3
Given, \(cos2\theta=sin\theta\)
\(\Rightarrow 1-2sin^2\theta=sin\theta\)
\(\Rightarrow (2sin \theta-1)(sin\theta+1)=0\)
\(\Rightarrow sin\theta =\frac{1}{2},sin\theta=-1\)
\(\Rightarrow \theta=\left \{ \frac{\pi}{6},\frac{5\pi}{6},\frac{3\pi}{2} \right \}\) as \(\theta\in (0,2\pi)\)
The number of solutions is 3.
Top Questions on Trigonometric Equations
If $ \theta \in [-2\pi,\ 2\pi] $, then the number of solutions of $$ 2\sqrt{2} \cos^2\theta + (2 - \sqrt{6}) \cos\theta - \sqrt{3} = 0 $$ is:
- JEE Main - 2025
- Mathematics
- Trigonometric Equations
- If \[ \lim_{x \to 0} \frac{\cos(2x) + a \cos(4x) - b}{x^4} \] is finite, then \( a + b = \) __.
- JEE Main - 2025
- Mathematics
- Trigonometric Equations
- Let $ A = \left\{ \theta \in [0, 2\pi] : \Re\left( \frac{2 \cos \theta + i \sin \theta}{\cos \theta - 3i \sin \theta} \right) = 0 \right\} $. Then $ \sum_{\theta \in A} \theta^2 $ is equal to:
- JEE Main - 2025
- Mathematics
- Trigonometric Equations
- If for $ \theta \in \left[ -\frac{\pi}{3}, 0 \right] $, the points $ (x, y) = \left( 3 \tan\left( \theta + \frac{\pi}{3} \right), 2 \tan\left( \theta + \frac{\pi}{6} \right) \right) $ lie on $ xy + \alpha x + \beta y + \gamma = 0 $, then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:
- JEE Main - 2025
- Mathematics
- Trigonometric Equations
- The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2\theta = \cos 2\theta \) and \( 2\cos^2\theta = 3\sin\theta \) is:
- JEE Main - 2025
- Mathematics
- Trigonometric Equations
Questions Asked in JEE Main exam
- The number of 6-letter words, with or without meaning, that can be formed using the letters of the word "MATHS" such that any letter that appears in the word must appear at least twice is:
- JEE Main - 2025
- Calculus
For \( \alpha, \beta, \gamma \in \mathbb{R} \), if \[ \lim_{x \to 0} \frac{x^2 \sin(\alpha x) + (\gamma - 1)e^{x^2}}{\sin(2x - \beta x)} = 3, \] then \( \beta + \gamma - \alpha \) is equal to:
- In \( I(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} \, dx \), where \( m, n > 0 \), then \( I(9, 14) + I(10, 13) \) is:
- JEE Main - 2025
- Functions
In the first configuration (1) as shown in the figure, four identical charges \( q_0 \) are kept at the corners A, B, C and D of square of side length \( a \). In the second configuration (2), the same charges are shifted to mid points C, E, H, and F of the square. If \( K = \frac{1}{4\pi \epsilon_0} \), the difference between the potential energies of configuration (2) and (1) is given by:
- JEE Main - 2025
- Electromagnetic Field (EMF)
If \( S \) and \( S' \) are the foci of the ellipse \[ \frac{x^2}{18} + \frac{y^2}{9} = 1 \] and \( P \) is a point on the ellipse, then \[ \min (SP \cdot S'P) + \max (SP \cdot S'P) \] is equal to:
- JEE Main - 2025
- Conic sections
Concepts Used:
Trigonometric Equations
Trigonometric equation is an equation involving one or more trigonometric ratios of unknown angles. It is expressed as ratios of sine(sin), cosine(cos), tangent(tan), cotangent(cot), secant(sec), cosecant(cosec) angles. For example, cos2 x + 5 sin x = 0 is a trigonometric equation. All possible values which satisfy the given trigonometric equation are called solutions of the given trigonometric equation.
A list of trigonometric equations and their solutions are given below:
| Trigonometrical equations | General Solutions |
| sin θ = 0 | θ = nπ |
| cos θ = 0 | θ = (nπ + π/2) |
| cos θ = 0 | θ = nπ |
| sin θ = 1 | θ = (2nπ + π/2) = (4n+1) π/2 |
| cos θ = 1 | θ = 2nπ |
| sin θ = sin α | θ = nπ + (-1)n α, where α ∈ [-π/2, π/2] |
| cos θ = cos α | θ = 2nπ ± α, where α ∈ (0, π] |
| tan θ = tan α | θ = nπ + α, where α ∈ (-π/2, π/2] |
| sin 2θ = sin 2α | θ = nπ ± α |
| cos 2θ = cos 2α | θ = nπ ± α |
| tan 2θ = tan 2α | θ = nπ ± α |