What is the general solution for 2*2^{sin(x)}=1 ?

The general solution for 2*2^{sin(x)}=1 is x=(3pi)/2+2pin

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Popular >

2*2^{sin(x)}=1

Solution

2 \cdot 2^{s i n (x)} = 1

Solution

x = \frac{3 π}{2} + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

2 \cdot 2^{s i n (x)} = 1

Apply exponent rules

2 \cdot 2^{s i n (x)} = 1

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{s i n (x)} \cdot 2 = 2^{s i n (x) + 1}

2^{s i n (x) + 1} = 1

f (x) = g (x)

, then

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

ln (2^{s i n (x) + 1}) = ln (1)

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln (2^{s i n (x) + 1}) = (sin (x) + 1) ln (2)

(sin (x) + 1) ln (2) = ln (1)

(sin (x) + 1) ln (2) = ln (1)

Solve

(sin (x) + 1) ln (2) = ln (1) : sin (x) = - 1

(sin (x) + 1) ln (2) = ln (1)

Divide both sides by

ln (2)

(sin (x) + 1) ln (2) = ln (1)

Divide both sides by

ln (2)

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

Simplify

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

Simplify

\frac{( sin ( x ) + 1 ) ln ( 2 )}{ln ( 2 )} : sin (x) + 1

\frac{( sin ( x ) + 1 ) ln ( 2 )}{ln ( 2 )}

Cancel the common factor:

ln (2)

= sin (x) + 1

Simplify

\frac{ln ( 1 )}{ln ( 2 )} : 0

\frac{ln ( 1 )}{ln ( 2 )}

Simplify

ln (1) : 0

ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 0

= \frac{0}{ln ( 2 )}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

sin (x) + 1 = 0

sin (x) + 1 = 0

sin (x) + 1 = 0

Move

1

to the right side

sin (x) + 1 = 0

Subtract

1

from both sides

sin (x) + 1 - 1 = 0 - 1

Simplify

sin (x) = - 1

sin (x) = - 1

sin (x) = - 1

General solutions for

sin (x) = - 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2*2^{sin(x)}=1 ?
The general solution for 2*2^{sin(x)}=1 is x=(3pi)/2+2pin