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Solve the trigonometric equation $\cos\left(x\right)^3-3\cos\left(x\right)=3\cos\left(x\right)\sin\left(x\right)$

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Solution

Final answer to the problem

$x=\frac{1}{2}\pi+2\pi n,\:x=\frac{3}{2}\pi+2\pi n,\:x=\frac{-1}{120}\pi+\frac{1}{360}\pi,\:x=\frac{-1}{120}\pi+\frac{-1}{360}\pi\:,\:\:n\in\Z$
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Step-by-step Solution

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Simplify $3\cos\left(x\right)\sin\left(x\right)$ using the trigonometric identity: $\sin(2x)=2\sin(x)\cos(x)$

$\cos\left(x\right)^3-3\cos\left(x\right)=\frac{3}{2}\sin\left(2x\right)$

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$\cos\left(x\right)^3-3\cos\left(x\right)=\frac{3}{2}\sin\left(2x\right)$

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Learn how to solve problems step by step online. Solve the trigonometric equation cos(x)^3-3cos(x)=3cos(x)sin(x). Simplify 3\cos\left(x\right)\sin\left(x\right) using the trigonometric identity: \sin(2x)=2\sin(x)\cos(x). Grouping all terms to the left side of the equation. Using the sine double-angle identity: \sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right). Factor the polynomial \cos\left(x\right)^3-3\cos\left(x\right)-3\sin\left(x\right)\cos\left(x\right) by it's greatest common factor (GCF): \cos\left(x\right).

Final answer to the problem

$x=\frac{1}{2}\pi+2\pi n,\:x=\frac{3}{2}\pi+2\pi n,\:x=\frac{-1}{120}\pi+\frac{1}{360}\pi,\:x=\frac{-1}{120}\pi+\frac{-1}{360}\pi\:,\:\:n\in\Z$

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Function Plot

Plotting: $\cos\left(x\right)^3-3\cos\left(x\right)-3\cos\left(x\right)\sin\left(x\right)$

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a
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f
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m
n
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x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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