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Popular >

sqrt(3)*sin(x/2)+cos(x)=1

Solution

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

Solution

x = 4 πn, x = 2 π + 4 πn, x = \frac{2 π}{3} + 4 πn, x = \frac{4 π}{3} + 4 πn

Degrees

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n, x = 12 0^{\circ} + 72 0^{\circ} n, x = 24 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

3 sin (\frac{x}{2}) + cos (x) - 1 = 0

Let:

u = \frac{x}{2}

3 sin (u) + cos (2 u) - 1 = 0

Rewrite using trig identities

- 1 + cos (2 u) + sin (u) 3

Use the Double Angle identity:

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

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

Simplify

= 3 sin (u) - 2 sin^{2} (u)

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

Solve by substitution

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

Let:

sin (u) = u

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

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 2, b = 3, c = 0

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

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

(3)^{2} - 4 (- 2) \cdot 0 = 3

(3)^{2} - 4 (- 2) \cdot 0

Apply rule

- (- a) = a

= (3)^{2} + 4 \cdot 2 \cdot 0

(3)^{2} = 3

(3)^{2}

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 \cdot 2 \cdot 0 = 0

4 \cdot 2 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 3 + 0

Add the numbers:

3 + 0 = 3

= 3

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

Separate the solutions

u_{1} = \frac{- 3 + 3}{2 ( - 2 )}, u_{2} = \frac{- 3 - 3}{2 ( - 2 )}

u = \frac{- 3 + 3}{2 ( - 2 )} : 0

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

Remove parentheses:

(- a) = - a

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

Add similar elements:

- 3 + 3 = 0

= \frac{0}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{0}{- 4}

Apply the fraction rule:

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

= - \frac{0}{4}

Apply rule

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

= - 0

= 0

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

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

Remove parentheses:

(- a) = - a

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

Add similar elements:

- 3 - 3 = - 23

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

Multiply the numbers:

2 \cdot 2 = 4

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

Apply the fraction rule:

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

= \frac{2 3}{4}

Cancel the common factor:

2

= \frac{3}{2}

The solutions to the quadratic equation are:

u = 0, u = \frac{3}{2}

Substitute back

u = sin (u)

sin (u) = 0, sin (u) = \frac{3}{2}

sin (u) = 0, sin (u) = \frac{3}{2}

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

General solutions for

sin (u) = 0

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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

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

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

General solutions for

sin (u) = \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}

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

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

Combine all the solutions

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

Substitute back

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplify

x = 4 πn

x = 4 πn

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

x = 2 π + 4 πn

x = 2 π + 4 πn

\frac{x}{2} = \frac{π}{3} + 2 πn : x = \frac{2 π}{3} + 4 πn

\frac{x}{2} = \frac{π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{3} + 2 \cdot 2 πn : \frac{2 π}{3} + 4 πn

2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= \frac{2 π}{3} + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

\frac{x}{2} = \frac{2 π}{3} + 2 πn : x = \frac{4 π}{3} + 4 πn

\frac{x}{2} = \frac{2 π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{2 π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn : \frac{4 π}{3} + 4 πn

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{2 π}{3} = \frac{4 π}{3}

2 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

x = 4 πn, x = 2 π + 4 πn, x = \frac{2 π}{3} + 4 πn, x = \frac{4 π}{3} + 4 πn

Graph

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