What is the general solution for e^{sin(x)}-2=0 ?

The general solution for e^{sin(x)}-2=0 is x=0.76584…+2pin,x=pi-0.76584…+2pin

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e^{sin(x)}-2=0

e^{s i n (x)} - 2 = 0

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

e^{s i n (x)} - 2 = 0

Add

2

to both sides

e^{s i n (x)} - 2 + 2 = 0 + 2

Simplify

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

Apply exponent rules

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

f (x) = g (x)

, then

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

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

Apply log rule:

ln (e^{a}) = a

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

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

(3 cos (x) - 6 sin (x))^{2} - 27 sin^{2} (x) = - 9

cos (2 x) = \frac{( 3 k - 2 )}{3}

sin^{2} (x) - sin (x) = 0, ([0, 2 π])

3 (cos (x) - sin (x)) = 0

4 tan (x) + 43 = 0

What is the general solution for e^{sin(x)}-2=0 ?
The general solution for e^{sin(x)}-2=0 is x=0.76584…+2pin,x=pi-0.76584…+2pin