What is the general solution for cos(x)+sqrt(3)sin(x)=2 ?

The general solution for cos(x)+sqrt(3)sin(x)=2 is x= pi/3+2pin

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cos(x)+sqrt(3)sin(x)=2

Solution

cos (x) + 3 sin (x) = 2

Solution

x = \frac{π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (x) + 3 sin (x) = 2

Subtract

3 sin (x)

from both sides

cos (x) = 2 - 3 sin (x)

Square both sides

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

Subtract

(2 - 3 sin (x))^{2}

from both sides

cos^{2} (x) - 4 + 43 sin (x) - 3 sin^{2} (x) = 0

Rewrite using trig identities

- 4 + cos^{2} (x) - 3 sin^{2} (x) + 4 sin (x) 3

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 4 + 1 - sin^{2} (x) - 3 sin^{2} (x) + 4 sin (x) 3

Simplify

- 4 + 1 - sin^{2} (x) - 3 sin^{2} (x) + 4 sin (x) 3 : 43 sin (x) - 4 sin^{2} (x) - 3

- 4 + 1 - sin^{2} (x) - 3 sin^{2} (x) + 4 sin (x) 3

Add similar elements:

- sin^{2} (x) - 3 sin^{2} (x) = - 4 sin^{2} (x)

= - 4 + 1 - 4 sin^{2} (x) + 43 sin (x)

Add/Subtract the numbers:

- 4 + 1 = - 3

= 43 sin (x) - 4 sin^{2} (x) - 3

= 43 sin (x) - 4 sin^{2} (x) - 3

- 3 - 4 sin^{2} (x) + 4 sin (x) 3 = 0

Solve by substitution

- 3 - 4 sin^{2} (x) + 4 sin (x) 3 = 0

Let:

sin (x) = u

- 3 - 4 u^{2} + 4 u 3 = 0

- 3 - 4 u^{2} + 4 u 3 = 0 : u = \frac{3}{2}

- 3 - 4 u^{2} + 4 u 3 = 0

Write in the standard form

a x^{2} + b x + c = 0

- 4 u^{2} + 43 u - 3 = 0

Solve with the quadratic formula

- 4 u^{2} + 43 u - 3 = 0

Quadratic Equation Formula:

For

a = - 4, b = 43, c = - 3

u_{1, 2} = \frac{- 4 3 \pm ( 4 3 ) ^{2} - 4 ( - 4 ) ( - 3 )}{2 ( - 4 )}

u_{1, 2} = \frac{- 4 3 \pm ( 4 3 ) ^{2} - 4 ( - 4 ) ( - 3 )}{2 ( - 4 )}

(43)^{2} - 4 (- 4) (- 3) = 0

(43)^{2} - 4 (- 4) (- 3)

Apply rule

- (- a) = a

= (43)^{2} - 4 \cdot 4 \cdot 3

(43)^{2} = 4^{2} \cdot 3

(43)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (3)^{2}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= 4^{2} \cdot 3

4 \cdot 4 \cdot 3 = 48

4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 48

= 4^{2} \cdot 3 - 48

4^{2} \cdot 3 = 48

4^{2} \cdot 3

4^{2} = 16

= 16 \cdot 3

Multiply the numbers:

16 \cdot 3 = 48

= 48

= 48 - 48

Subtract the numbers:

48 - 48 = 0

= 0

u_{1, 2} = \frac{- 4 3 \pm 0}{2 ( - 4 )}

u = \frac{- 4 3}{2 ( - 4 )}

\frac{- 4 3}{2 ( - 4 )} = \frac{3}{2}

\frac{- 4 3}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 3}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4 3}{- 8}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4 3}{8}

Cancel the common factor:

4

= \frac{3}{2}

u = \frac{3}{2}

The solution to the quadratic equation is:

u = \frac{3}{2}

Substitute back

u = sin (x)

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

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

sin (x) = \frac{3}{2} : x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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

General solutions for

sin (x) = \frac{3}{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}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

cos (x) + 3 sin (x) = 2

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

Check the solution

\frac{π}{3} + 2 πn :

True

\frac{π}{3} + 2 πn

Plug in

n = 1

\frac{π}{3} + 2 π 1

For

cos (x) + 3 sin (x) = 2

plug in

x = \frac{π}{3} + 2 π 1

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

Refine

2 = 2

\Rightarrow True

Check the solution

\frac{2 π}{3} + 2 πn :

False

\frac{2 π}{3} + 2 πn

Plug in

n = 1

\frac{2 π}{3} + 2 π 1

For

cos (x) + 3 sin (x) = 2

plug in

x = \frac{2 π}{3} + 2 π 1

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

Refine

1 = 2

\Rightarrow False

x = \frac{π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x)+sqrt(3)sin(x)=2 ?
The general solution for cos(x)+sqrt(3)sin(x)=2 is x= pi/3+2pin