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800cos(x)+800cos^2(x)-800sin^2(x)=0

Solution

800 cos (x) + 800 cos^{2} (x) - 800 sin^{2} (x) = 0

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

800 cos (x) + 800 cos^{2} (x) - 800 sin^{2} (x) = 0

Rewrite using trig identities

800 cos (x) + 800 cos^{2} (x) - 800 sin^{2} (x)

Use the Pythagorean identity:

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

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

= 800 cos (x) + 800 cos^{2} (x) - 800 (1 - cos^{2} (x))

Simplify

800 cos (x) + 800 cos^{2} (x) - 800 (1 - cos^{2} (x)) : 800 cos (x) + 1600 cos^{2} (x) - 800

800 cos (x) + 800 cos^{2} (x) - 800 (1 - cos^{2} (x))

Expand

- 800 (1 - cos^{2} (x)) : - 800 + 800 cos^{2} (x)

- 800 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 800, b = 1, c = cos^{2} (x)

= - 800 \cdot 1 - (- 800) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 800 \cdot 1 + 800 cos^{2} (x)

Multiply the numbers:

800 \cdot 1 = 800

= - 800 + 800 cos^{2} (x)

= 800 cos (x) + 800 cos^{2} (x) - 800 + 800 cos^{2} (x)

Simplify

800 cos (x) + 800 cos^{2} (x) - 800 + 800 cos^{2} (x) : 800 cos (x) + 1600 cos^{2} (x) - 800

800 cos (x) + 800 cos^{2} (x) - 800 + 800 cos^{2} (x)

Group like terms

= 800 cos (x) + 800 cos^{2} (x) + 800 cos^{2} (x) - 800

Add similar elements:

800 cos^{2} (x) + 800 cos^{2} (x) = 1600 cos^{2} (x)

= 800 cos (x) + 1600 cos^{2} (x) - 800

= 800 cos (x) + 1600 cos^{2} (x) - 800

= 800 cos (x) + 1600 cos^{2} (x) - 800

- 800 + 1600 cos^{2} (x) + 800 cos (x) = 0

Solve by substitution

- 800 + 1600 cos^{2} (x) + 800 cos (x) = 0

Let:

cos (x) = u

- 800 + 1600 u^{2} + 800 u = 0

- 800 + 1600 u^{2} + 800 u = 0 : u = \frac{1}{2}, u = - 1

- 800 + 1600 u^{2} + 800 u = 0

Divide both sides by

1600

- \frac{800}{1600} + \frac{1600 u ^{2}}{1600} + \frac{800 u}{1600} = \frac{0}{1600}

Write in the standard form

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

u^{2} + \frac{1 \cdot u}{2} - \frac{1}{2} = 0

Solve with the quadratic formula

u^{2} + \frac{1 \cdot u}{2} - \frac{1}{2} = 0

Quadratic Equation Formula:

For

a = 1, b = \frac{1}{2}, c = - \frac{1}{2}

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

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

(\frac{1}{2})^{2} - 4 \cdot 1 \cdot (- \frac{1}{2}) = \frac{3}{2}

(\frac{1}{2})^{2} - 4 \cdot 1 \cdot (- \frac{1}{2})

Apply rule

- (- a) = a

= (\frac{1}{2})^{2} + 4 \cdot 1 \cdot \frac{1}{2}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

4 \cdot 1 \cdot \frac{1}{2} = 2

4 \cdot 1 \cdot \frac{1}{2}

Multiply fractions:

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

= 1 \cdot \frac{1 \cdot 4}{2}

\frac{1 \cdot 4}{2} = 2

\frac{1 \cdot 4}{2}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

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

2^{2} = 4

= \frac{1}{4} + 2

Join

\frac{1}{4} + 2 : \frac{9}{4}

\frac{1}{4} + 2

Convert element to fraction:

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

= \frac{2 \cdot 4}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{2 \cdot 4 + 1}{4}

2 \cdot 4 + 1 = 9

2 \cdot 4 + 1

Multiply the numbers:

2 \cdot 4 = 8

= 8 + 1

Add the numbers:

8 + 1 = 9

= 9

= \frac{9}{4}

= \frac{9}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{9}{2}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{2}

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

Separate the solutions

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

u = \frac{- \frac{1}{2} + \frac{3}{2}}{2 \cdot 1} : \frac{1}{2}

\frac{- \frac{1}{2} + \frac{3}{2}}{2 \cdot 1}

Combine the fractions

- \frac{1}{2} + \frac{3}{2} : 1

Apply rule

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

= \frac{- 1 + 3}{2}

Add/Subtract the numbers:

- 1 + 3 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= \frac{1}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{1}{2}

u = \frac{- \frac{1}{2} - \frac{3}{2}}{2 \cdot 1} : - 1

\frac{- \frac{1}{2} - \frac{3}{2}}{2 \cdot 1}

Combine the fractions

- \frac{1}{2} - \frac{3}{2} : - 2

Apply rule

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

= \frac{- 1 - 3}{2}

Subtract the numbers:

- 1 - 3 = - 4

= \frac{- 4}{2}

Apply the fraction rule:

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

= - \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2

= \frac{- 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

General solutions for

cos (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}

x = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

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