What is the general solution for sqrt(4sin(x)+7)=3 ?

The general solution for sqrt(4sin(x)+7)=3 is x= pi/6+2pin,x=(5pi)/6+2pin

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Popular >

sqrt(4sin(x)+7)=3

Solution

4 sin (x) + 7 = 3

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

4 sin (x) + 7 = 3

Square both sides:

4 sin (x) + 7 = 9

4 sin (x) + 7 = 3

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

Expand

(4 sin (x) + 7)^{2} : 4 sin (x) + 7

(4 sin (x) + 7)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((4 sin (x) + 7)^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (4 sin (x) + 7)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 4 sin (x) + 7

Expand

3^{2} : 9

3^{2}

3^{2} = 9

= 9

4 sin (x) + 7 = 9

4 sin (x) + 7 = 9

Solve

4 sin (x) + 7 = 9 : sin (x) = \frac{1}{2}

4 sin (x) + 7 = 9

Move

7

to the right side

4 sin (x) + 7 = 9

Subtract

7

from both sides

4 sin (x) + 7 - 7 = 9 - 7

Simplify

4 sin (x) = 2

4 sin (x) = 2

Divide both sides by

4

4 sin (x) = 2

Divide both sides by

4

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

Simplify

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

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

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

Verify Solutions:

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

True

Check the solutions by plugging them into

4 sin (x) + 7 = 3

Remove the ones that don't agree with the equation.

Plug in

sin (x) = \frac{1}{2} :

True

4 (\frac{1}{2}) + 7 = 3

4 (\frac{1}{2}) + 7 = 3

4 (\frac{1}{2}) + 7

Remove parentheses:

(a) = a

= 4 \cdot \frac{1}{2} + 7

4 \cdot \frac{1}{2} = 2

4 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 4}{2}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 2 + 7

Add the numbers:

2 + 7 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

3 = 3

True

The solution is

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

General solutions for

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

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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Popular Examples

1-cos(4x)=sin(2x)

1 - cos (4 x) = sin (2 x)

tan(θ)= 2/1

tan (θ) = \frac{2}{1}

cos(2x)=sin^2(x)

cos (2 x) = sin^{2} (x)

sec^2(x)-3tan(x)=5

sec^{2} (x) - 3 tan (x) = 5

sin(x)=0,4

sin (x) = 0, 4

Frequently Asked Questions (FAQ)

What is the general solution for sqrt(4sin(x)+7)=3 ?
The general solution for sqrt(4sin(x)+7)=3 is x= pi/6+2pin,x=(5pi)/6+2pin