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Popular Trigonometry >

21+18cos(x)=16(1-cos^2(x))

Solution

21 + 18 cos (x) = 16 (1 - cos^{2} (x))

Solution

x = 2.24592 \dots + 2 πn, x = - 2.24592 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Degrees

x = 128.68218 \dots^{\circ} + 36 0^{\circ} n, x = - 128.68218 \dots^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

21 + 18 cos (x) = 16 (1 - cos^{2} (x))

Solve by substitution

21 + 18 cos (x) = 16 (1 - cos^{2} (x))

Let:

cos (x) = u

21 + 18 u = 16 (1 - u^{2})

21 + 18 u = 16 (1 - u^{2}) : u = - \frac{5}{8}, u = - \frac{1}{2}

21 + 18 u = 16 (1 - u^{2})

Expand

16 (1 - u^{2}) : 16 - 16 u^{2}

16 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 16, b = 1, c = u^{2}

= 16 \cdot 1 - 16 u^{2}

Multiply the numbers:

16 \cdot 1 = 16

= 16 - 16 u^{2}

21 + 18 u = 16 - 16 u^{2}

Switch sides

16 - 16 u^{2} = 21 + 18 u

Move

18 u

to the left side

16 - 16 u^{2} = 21 + 18 u

Subtract

18 u

from both sides

16 - 16 u^{2} - 18 u = 21 + 18 u - 18 u

Simplify

16 - 16 u^{2} - 18 u = 21

16 - 16 u^{2} - 18 u = 21

Move

21

to the left side

16 - 16 u^{2} - 18 u = 21

Subtract

21

from both sides

16 - 16 u^{2} - 18 u - 21 = 21 - 21

Simplify

- 16 u^{2} - 18 u - 5 = 0

- 16 u^{2} - 18 u - 5 = 0

Solve with the quadratic formula

- 16 u^{2} - 18 u - 5 = 0

Quadratic Equation Formula:

For

a = - 16, b = - 18, c = - 5

u_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 ( - 16 ) ( - 5 )}{2 ( - 16 )}

u_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 ( - 16 ) ( - 5 )}{2 ( - 16 )}

(- 18)^{2} - 4 (- 16) (- 5) = 2

(- 18)^{2} - 4 (- 16) (- 5)

Apply rule

- (- a) = a

= (- 18)^{2} - 4 \cdot 16 \cdot 5

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 18)^{2} = 1 8^{2}

= 1 8^{2} - 4 \cdot 16 \cdot 5

Multiply the numbers:

4 \cdot 16 \cdot 5 = 320

= 1 8^{2} - 320

1 8^{2} = 324

= 324 - 320

Subtract the numbers:

324 - 320 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u_{1, 2} = \frac{- ( - 18 ) \pm 2}{2 ( - 16 )}

Separate the solutions

u_{1} = \frac{- ( - 18 ) + 2}{2 ( - 16 )}, u_{2} = \frac{- ( - 18 ) - 2}{2 ( - 16 )}

u = \frac{- ( - 18 ) + 2}{2 ( - 16 )} : - \frac{5}{8}

\frac{- ( - 18 ) + 2}{2 ( - 16 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{18 + 2}{- 2 \cdot 16}

Add the numbers:

18 + 2 = 20

= \frac{20}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{20}{- 32}

Apply the fraction rule:

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

= - \frac{20}{32}

Cancel the common factor:

4

= - \frac{5}{8}

u = \frac{- ( - 18 ) - 2}{2 ( - 16 )} : - \frac{1}{2}

\frac{- ( - 18 ) - 2}{2 ( - 16 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{18 - 2}{- 2 \cdot 16}

Subtract the numbers:

18 - 2 = 16

= \frac{16}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{16}{- 32}

Apply the fraction rule:

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

= - \frac{16}{32}

Cancel the common factor:

16

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = - \frac{5}{8}, u = - \frac{1}{2}

Substitute back

u = cos (x)

cos (x) = - \frac{5}{8}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{5}{8}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{5}{8} : x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

cos (x) = - \frac{5}{8}

Apply trig inverse properties

cos (x) = - \frac{5}{8}

General solutions for

cos (x) = - \frac{5}{8}

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

x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

General solutions for

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

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}

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

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

Combine all the solutions

x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Show solutions in decimal form

x = 2.24592 \dots + 2 πn, x = - 2.24592 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Graph

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