Solve the equation sin- -sin3- -sin5- -0

sinθ+sin3θ+sin5θ=0 ⇒(sinθ+sin5θ)+sin3θ=0 ⇒ 2sin(θ+5θ2)cos(θ 5θ2)+sin3θ=0 ⇒ 2sin3θcos2θ+sin3θ=0 ⇒sin3θ(2cos2θ+1)=0 ⇒sin3θ=0,cos2θ= ...

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Byju's Answer

Standard XII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

Solve the equ...

Question

Solve the equation
$s i n θ + s i n 3 θ + s i n 5 θ = 0$

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Solution

$sin θ + sin 3 θ + sin 5 θ = 0$
$\Rightarrow (sin θ + sin 5 θ) + sin 3 θ = 0$
$\Rightarrow 2 sin (\frac{θ + 5 θ}{2}) cos (\frac{θ - 5 θ}{2}) + sin 3 θ = 0$
$\Rightarrow 2 sin 3 θ cos 2 θ + sin 3 θ = 0$
$\Rightarrow sin 3 θ (2 cos 2 θ + 1) = 0$
$\Rightarrow sin 3 θ = 0, cos 2 θ = \frac{- 1}{2}$
$\Rightarrow 3 θ = n π, 2 θ = 2 n π + π - \frac{π}{3}$
$∴ θ = \frac{n π}{3}, n π + \frac{π}{3}$
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Similar questions

Q. Find the number of roots of the equation

sin θ + sin 5 θ = sin 3 θ

[0, π]

$Solve the following equations : (i) c o s θ + c p s 2 θ + c o s 3 θ = 0 (i i) c o s θ + c o s 3 θ - c o s 2 θ = 0 (i i i) s i n θ + s i n 5 θ = s i n 3 θ (i v) c o s θ c o s 2 θ c o s 3 θ = \frac{1}{4} (v) c o s θ + s i n θ = c o s 2 θ + s i n 2 θ (v i) s i n θ + s i n 2 θ + s i n 3 θ = 0 (v i i) s i n θ + s i n 2 θ + s i n 3 θ + s i n 4 θ = 0 (v i i i) s i n 3 θ - s i n θ = 4 c o s^{2} θ - 2 (i x) s i n 2 θ - s i n 4 θ + s i n 6 θ = 0$

Q. Find all the values of

θ

satisfying the equation :

sin θ + sin 5 θ = sin 3 θ

such that

0 \leq θ \leq π

Q. If the value of

sin θ + sin 3 θ + sin 5 θ + . . . + sin (2 n - 1) θ = \frac{{sin}^{a} n θ}{sin θ}

Find

a .

\frac{sin 3 θ + 2 sin 5 θ + sin 7 θ}{sin θ + 2 sin 3 θ + sin 5 θ} = cos 2 θ + sin 2 θ . cot 3 θ

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Solve the equationsinθ+sin3θ+sin5θ=0

sinθ+sin3θ+sin5θ=0⇒(sinθ+sin5θ)+sin3θ=0⇒2sin(θ+5θ2)cos(θ−5θ2)+sin3θ=0⇒2sin3θcos2θ+sin3θ=0⇒sin3θ(2cos2θ+1)=0⇒sin3θ=0,cos2θ=−12⇒3θ=nπ,2θ=2nπ+π−π3∴θ=nπ3,nπ+π3'

Solve the equation
$s i n θ + s i n 3 θ + s i n 5 θ = 0$