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Solve the trigonometric equation $\sin\left(x\right)+\frac{-\sqrt{2}}{2}=0$

Step-by-step Solution

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Final Answer

$x=\frac{1}{4}\pi+2\pi n,\:x=\frac{3}{4}\pi+2\pi n\:,\:\:n\in\Z$
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Step-by-step Solution

Specify the solving method

1

Calculate the square root of $2$

$\sin\left(x\right)+\frac{-\sqrt{2}}{2}=0$
2

Multiply $-1$ times $\sqrt{2}$

$\sin\left(x\right)-\frac{\sqrt{2}}{2}=0$
3

Divide $-\sqrt{2}$ by $2$

$\sin\left(x\right)-\frac{\sqrt{2}}{2}=0$

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-\frac{\sqrt{2}}{2}$ from both sides of the equation

$\sin\left(x\right)=0+\frac{\sqrt{2}}{2}$

$x+0=x$, where $x$ is any expression

$\sin\left(x\right)=\frac{\sqrt{2}}{2}$
4

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-\frac{\sqrt{2}}{2}$ from both sides of the equation

$\sin\left(x\right)=\frac{\sqrt{2}}{2}$
5

The angles where the function $\sin\left(x\right)$ is $\frac{\sqrt{2}}{2}$ are

$x=45^{\circ}+360^{\circ}n,\:x=135^{\circ}+360^{\circ}n$
6

The angles expressed in radians in the same order are equal to

$x=\frac{1}{4}\pi+2\pi n,\:x=\frac{3}{4}\pi+2\pi n\:,\:\:n\in\Z$

Final Answer

$x=\frac{1}{4}\pi+2\pi n,\:x=\frac{3}{4}\pi+2\pi n\:,\:\:n\in\Z$

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0
a
b
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f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
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

How to improve your answer:

Main Topic: Implicit Differentiation

Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. For differentiating an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y(x) and then differentiate. Instead, one can differentiate R(x, y) with respect to x and y and then solve a linear equation in dy/dx for getting explicitly the derivative in terms of x and y.

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