What is the general solution for 7=e^{2sin(x)}+3 ?

The general solution for 7=e^{2sin(x)}+3 is x=0.76584…+2pin,x=pi-0.76584…+2pin

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Popular >

7=e^{2sin(x)}+3

Solution

7 = e^{2 s i n (x)} + 3

Solution

x = 0.76584 \dots + 2 πn, x = π - 0.76584 \dots + 2 πn

Degrees

x = 43.87975 \dots^{\circ} + 36 0^{\circ} n, x = 136.12024 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 = e^{2 s i n (x)} + 3

Switch sides

e^{2 s i n (x)} + 3 = 7

Move

3

to the right side

e^{2 s i n (x)} + 3 = 7

Subtract

3

from both sides

e^{2 s i n (x)} + 3 - 3 = 7 - 3

Simplify

e^{2 s i n (x)} = 4

e^{2 s i n (x)} = 4

Apply exponent rules

e^{2 s i n (x)} = 4

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{2 s i n (x)}) = ln (4)

Apply log rule:

ln (e^{a}) = a

ln (e^{2 s i n (x)}) = 2 sin (x)

2 sin (x) = ln (4)

2 sin (x) = ln (4)

Solve

2 sin (x) = ln (4) : sin (x) = ln (2)

2 sin (x) = ln (4)

Divide both sides by

2

2 sin (x) = ln (4)

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{ln ( 4 )}{2}

Simplify

\frac{2 sin ( x )}{2} = \frac{ln ( 4 )}{2}

Simplify

\frac{2 sin ( x )}{2} : sin (x)

\frac{2 sin ( x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= sin (x)

Simplify

\frac{ln ( 4 )}{2} : ln (2)

\frac{ln ( 4 )}{2}

Simplify

ln (4) : 2 ln (2)

ln (4)

Rewrite

4

in power-base form:

4 = 2^{2}

= ln (2^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (2^{2}) = 2 ln (2)

= 2 ln (2)

= \frac{2 ln ( 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= ln (2)

sin (x) = ln (2)

sin (x) = ln (2)

sin (x) = ln (2)

sin (x) = ln (2)

Apply trig inverse properties

sin (x) = ln (2)

General solutions for

sin (x) = ln (2)

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (ln (2)) + 2 πn, x = π - arcsin (ln (2)) + 2 πn

x = arcsin (ln (2)) + 2 πn, x = π - arcsin (ln (2)) + 2 πn

Show solutions in decimal form

x = 0.76584 \dots + 2 πn, x = π - 0.76584 \dots + 2 πn

Popular Examples

sin(u)=-0.9

sin (u) = - 0.9

-3/2 =-1/2 sec(x)

- \frac{3}{2} = - \frac{1}{2} sec (x)

sin(3x+1)= 1/2

sin (3 x + 1) = \frac{1}{2}

cos(x)=(13^2+18^2-28^2)/(2*13*18)

cos (x) = \frac{1 3 ^{2} + 1 8 ^{2} - 2 8 ^{2}}{2 \cdot 13 \cdot 18}

0.16085=sin((2pi)/(365)(x-114))

0.16085 = sin (\frac{2 π}{365} (x - 114))

Frequently Asked Questions (FAQ)

What is the general solution for 7=e^{2sin(x)}+3 ?
The general solution for 7=e^{2sin(x)}+3 is x=0.76584…+2pin,x=pi-0.76584…+2pin