What is the general solution for 2sin(30+x)=3cos(x) ?

The general solution for 2sin(30+x)=3cos(x) is x=0.85707…+180n

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2sin(30+x)=3cos(x)

Solution

2 sin (3 0^{\circ} + x) = 3 cos (x)

Solution

x = 0.85707 \dots + 18 0^{\circ} n

Radians

x = 0.85707 \dots + πn

Solution steps

2 sin (3 0^{\circ} + x) = 3 cos (x)

Rewrite using trig identities

2 sin (3 0^{\circ} + x) = 3 cos (x)

Rewrite using trig identities

sin (3 0^{\circ} + x)

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (3 0^{\circ}) cos (x) + cos (3 0^{\circ}) sin (x)

Simplify

sin (3 0^{\circ}) cos (x) + cos (3 0^{\circ}) sin (x) : \frac{1}{2} cos (x) + \frac{3}{2} sin (x)

sin (3 0^{\circ}) cos (x) + cos (3 0^{\circ}) sin (x)

Simplify

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

Simplify

cos (3 0^{\circ}) : \frac{3}{2}

cos (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

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

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

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

Simplify

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = \frac{1}{2} cos (x), c = \frac{3}{2} sin (x)

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

Simplify

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

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

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

2 \cdot \frac{1}{2} cos (x)

Multiply fractions:

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

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

Cancel the common factor:

2

= cos (x) \cdot 1

Multiply:

cos (x) \cdot 1 = cos (x)

= cos (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= sin (x) 3

= cos (x) + 3 sin (x)

= cos (x) + 3 sin (x)

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

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

Subtract

3 cos (x)

from both sides

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

Rewrite using trig identities

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

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

- 2 + 3 tan (x) = 0

- 2 + 3 tan (x) = 0

Move

2

to the right side

- 2 + 3 tan (x) = 0

Add

2

to both sides

- 2 + 3 tan (x) + 2 = 0 + 2

Simplify

3 tan (x) = 2

3 tan (x) = 2

Divide both sides by

3

3 tan (x) = 2

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

Simplify

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

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

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

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

Apply trig inverse properties

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

General solutions for

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

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

x = arctan (\frac{2 3}{3}) + 18 0^{\circ} n

x = arctan (\frac{2 3}{3}) + 18 0^{\circ} n

Show solutions in decimal form

x = 0.85707 \dots + 18 0^{\circ} n

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2 arccos (x) = π

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(30+x)=3cos(x) ?
The general solution for 2sin(30+x)=3cos(x) is x=0.85707…+180n