If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.

प्रश्न

If 2sin²θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.

बेरीज

उत्तर

2sin²θ = 3cosθ

We know that,

sin²θ = 1 – cos²θ

Given that,

2sin²θ = 3 cosθ

2 – 2cos²θ = 3cosθ

2cos²θ + 3cosθ – 2 = 0

(cosθ + 2)(2cosθ – 1) = 0

Therefore,

cosθ = `1/2 = cos pi/3`

θ = `pi/3` or `2π – pi/3`

θ = `pi/3, (5pi)/3`

Therefore, 2(1 – cos²θ) = 3cosθ

⇒ 2 – 2cos²θ = 3cosθ

⇒ 2cos²θ + 3cosθ – 2 = 0

⇒ 2cos²θ + 4cosθ – cosθ – 2 = 0

⇒ 2cosθ(cosθ + 2) + 1(cosθ + 2) = 0

⇒ (2cosθ + 1)(cosθ + 2) = 0

Since, cosθ ∈ [–1, 1], for any value θ.

So, cosθ ≠ –2

Therefore,

2cosθ – 1 = 0

⇒ cosθ = `1/2`

= `pi/3` or `2π – pi/3`

θ = `π/3, (5pi)/3`

shaalaa.com

Trigonometric Equations

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 18 Q 17 Q 19

पाठ 3: Trigonometric Functions - Exercise [पृष्ठ ५४]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 11

पाठ 3 Trigonometric Functions
Exercise | Q 18 | पृष्ठ ५४

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

view
व्हिडिओ ट्यूटोरियल For All Subjects
Trigonometric Equations

video tutorial00:19:05

संबंधित प्रश्‍न

Find the general solution of cosec x = –2

Find the general solution of the equation sin 2x + cos x = 0

Prove the:
\[ \sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}} = - \frac{2}{\cos x},\text{ where }\frac{\pi}{2} < x < \pi\]

If \[T_n = \sin^n x + \cos^n x\], prove that \[2 T_6 - 3 T_4 + 1 = 0\]

If \[T_n = \sin^n x + \cos^n x\], prove that \[6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0\]

Prove that:

\[3\sin\frac{\pi}{6}\sec\frac{\pi}{3} - 4\sin\frac{5\pi}{6}\cot\frac{\pi}{4} = 1\]

Prove that

\[\frac{cosec(90^\circ + x) + \cot(450^\circ + x)}{cosec(90^\circ - x) + \tan(180^\circ - x)} + \frac{\tan(180^\circ + x) + \sec(180^\circ - x)}{\tan(360^\circ + x) - \sec( - x)} = 2\]

Find x from the following equations:
\[cosec\left( \frac{\pi}{2} + \theta \right) + x \cos \theta \cot\left( \frac{\pi}{2} + \theta \right) = \sin\left( \frac{\pi}{2} + \theta \right)\]

Find x from the following equations:
\[x \cot\left( \frac{\pi}{2} + \theta \right) + \tan\left( \frac{\pi}{2} + \theta \right)\sin \theta + cosec\left( \frac{\pi}{2} + \theta \right) = 0\]

Prove that:
\[\sin \frac{13\pi}{3}\sin\frac{2\pi}{3} + \cos\frac{4\pi}{3}\sin\frac{13\pi}{6} = \frac{1}{2}\]

If sec \[x = x + \frac{1}{4x}\], then sec x + tan x =

If tan x + sec x = \[\sqrt{3}\], 0 < x < π, then x is equal to

\[\sec^2 x = \frac{4xy}{(x + y )^2}\] is true if and only if

Find the general solution of the following equation:

\[\sin x = \frac{1}{2}\]

Find the general solution of the following equation:

\[\sqrt{3} \sec x = 2\]

Find the general solution of the following equation:

\[\cos 3x = \frac{1}{2}\]

Find the general solution of the following equation:

\[\tan mx + \cot nx = 0\]