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Popular >

25cos^2(3x)+10cos(3x)=15

Solution

25 cos^{2} (3 x) + 10 cos (3 x) = 15

Solution

x = \frac{0.92729 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.92729 \dots}{3} + \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

Degrees

x = 17.71003 \dots^{\circ} + 12 0^{\circ} n, x = 102.28996 \dots^{\circ} + 12 0^{\circ} n, x = 6 0^{\circ} + 12 0^{\circ} n

Solution steps

25 cos^{2} (3 x) + 10 cos (3 x) = 15

Solve by substitution

25 cos^{2} (3 x) + 10 cos (3 x) = 15

Let:

cos (3 x) = u

25 u^{2} + 10 u = 15

25 u^{2} + 10 u = 15 : u = \frac{3}{5}, u = - 1

25 u^{2} + 10 u = 15

Move

15

to the left side

25 u^{2} + 10 u = 15

Subtract

15

from both sides

25 u^{2} + 10 u - 15 = 15 - 15

Simplify

25 u^{2} + 10 u - 15 = 0

25 u^{2} + 10 u - 15 = 0

Solve with the quadratic formula

25 u^{2} + 10 u - 15 = 0

Quadratic Equation Formula:

For

a = 25, b = 10, c = - 15

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 25 ( - 15 )}{2 \cdot 25}

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 25 ( - 15 )}{2 \cdot 25}

1 0^{2} - 4 \cdot 25 (- 15) = 40

1 0^{2} - 4 \cdot 25 (- 15)

Apply rule

- (- a) = a

= 1 0^{2} + 4 \cdot 25 \cdot 15

Multiply the numbers:

4 \cdot 25 \cdot 15 = 1500

= 1 0^{2} + 1500

1 0^{2} = 100

= 100 + 1500

Add the numbers:

100 + 1500 = 1600

= 1600

Factor the number:

1600 = 4 0^{2}

= 4 0^{2}

Apply radical rule:

4 0^{2} = 40

= 40

u_{1, 2} = \frac{- 10 \pm 40}{2 \cdot 25}

Separate the solutions

u_{1} = \frac{- 10 + 40}{2 \cdot 25}, u_{2} = \frac{- 10 - 40}{2 \cdot 25}

u = \frac{- 10 + 40}{2 \cdot 25} : \frac{3}{5}

\frac{- 10 + 40}{2 \cdot 25}

Add/Subtract the numbers:

- 10 + 40 = 30

= \frac{30}{2 \cdot 25}

Multiply the numbers:

2 \cdot 25 = 50

= \frac{30}{50}

Cancel the common factor:

10

= \frac{3}{5}

u = \frac{- 10 - 40}{2 \cdot 25} : - 1

\frac{- 10 - 40}{2 \cdot 25}

Subtract the numbers:

- 10 - 40 = - 50

= \frac{- 50}{2 \cdot 25}

Multiply the numbers:

2 \cdot 25 = 50

= \frac{- 50}{50}

Apply the fraction rule:

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

= - \frac{50}{50}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = \frac{3}{5}, u = - 1

Substitute back

u = cos (3 x)

cos (3 x) = \frac{3}{5}, cos (3 x) = - 1

cos (3 x) = \frac{3}{5}, cos (3 x) = - 1

cos (3 x) = \frac{3}{5} : x = \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

cos (3 x) = \frac{3}{5}

Apply trig inverse properties

cos (3 x) = \frac{3}{5}

General solutions for

cos (3 x) = \frac{3}{5}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

3 x = arccos (\frac{3}{5}) + 2 πn, 3 x = 2 π - arccos (\frac{3}{5}) + 2 πn

3 x = arccos (\frac{3}{5}) + 2 πn, 3 x = 2 π - arccos (\frac{3}{5}) + 2 πn

Solve

3 x = arccos (\frac{3}{5}) + 2 πn : x = \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

3 x = arccos (\frac{3}{5}) + 2 πn

Divide both sides by

3

3 x = arccos (\frac{3}{5}) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

Simplify

x = \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

Solve

3 x = 2 π - arccos (\frac{3}{5}) + 2 πn : x = \frac{2 π}{3} - \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

3 x = 2 π - arccos (\frac{3}{5}) + 2 πn

Divide both sides by

3

3 x = 2 π - arccos (\frac{3}{5}) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 π}{3} - \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

Simplify

x = \frac{2 π}{3} - \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} - \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

cos (3 x) = - 1 : x = \frac{π}{3} + \frac{2 πn}{3}

cos (3 x) = - 1

General solutions for

cos (3 x) = - 1

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

3 x = π + 2 πn

3 x = π + 2 πn

Solve

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

3 x = π + 2 πn

Divide both sides by

3

3 x = π + 2 πn

Divide both sides by

3

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

Simplify

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{3}{5} )}{3} + \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

Show solutions in decimal form

x = \frac{0.92729 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.92729 \dots}{3} + \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

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