Solve 2 cos^2(\theta ) - cos(\theta ) - 1 = 0

Question

Solve the equation

θ = \frac{2 kπ}{3}, k \in Z

Alternative Form

θ = 12 0^{\circ} k, k \in Z

Evaluate

2 cos^{2} (θ) - cos (θ) - 1 = 0

Factor the expression

More Steps

Calculate

2 cos^{2} (θ) - cos (θ) - 1

Rewrite the expression

2 cos^{2} (θ) + (1 - 2) cos (θ) - 1

Calculate

2 cos^{2} (θ) + cos (θ) - 2 cos (θ) - 1

Rewrite the expression

cos (θ) \times 2 cos (θ) + cos (θ) - 2 cos (θ) - 1

Factor out cos (θ) from the expression

cos (θ) (2 cos (θ) + 1) - 2 cos (θ) - 1

Factor out - 1 from the expression

cos (θ) (2 cos (θ) + 1) - (2 cos (θ) + 1)

Factor out 2 cos (θ) + 1 from the expression

(cos (θ) - 1) (2 cos (θ) + 1)

(cos (θ) - 1) (2 cos (θ) + 1) = 0

Separate the equation into 2 possible cases

cos (θ) - 1 = 0 2 cos (θ) + 1 = 0

Solve the equation

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Evaluate

cos (θ) - 1 = 0

Move the constant to the right-hand side and change its sign

cos (θ) = 0 + 1

Removing 0 doesn't change the value,so remove it from the expression

cos (θ) = 1

Use the inverse trigonometric function

θ = arccos (1)

Calculate

θ = 0

Add the period of 2 kπ, k \in Z to find all solutions

θ = 2 kπ, k \in Z

θ = 2 kπ, k \in Z 2 cos (θ) + 1 = 0

Solve the equation

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Evaluate

2 cos (θ) + 1 = 0

Move the constant to the right-hand side and change its sign

2 cos (θ) = 0 - 1

Removing 0 doesn't change the value,so remove it from the expression

2 cos (θ) = - 1

Divide both sides

\frac{2 cos ( θ )}{2} = \frac{- 1}{2}

Divide the numbers

cos (θ) = \frac{- 1}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

cos (θ) = - \frac{1}{2}

Use the inverse trigonometric function

θ = arccos (- \frac{1}{2})

Calculate

θ = \frac{2 π}{3} θ = \frac{4 π}{3}

Add the period of 2 kπ, k \in Z to find all solutions

θ = \frac{2 π}{3} + 2 kπ, k \in Z θ = \frac{4 π}{3} + 2 kπ, k \in Z

Find the union

θ = {\frac{2 π}{3} + 2 kπ \frac{4 π}{3} + 2 kπ, k \in Z

θ = 2 kπ, k \in Z θ = {\frac{2 π}{3} + 2 kπ \frac{4 π}{3} + 2 kπ, k \in Z

Solution

θ = \frac{2 kπ}{3}, k \in Z

Alternative Form

θ = 12 0^{\circ} k, k \in Z

Show Solution

Rewrite the equation

3 x^{2} y^{2} = y^{4}

Evaluate

2 cos^{2} (θ) - cos (θ) - 1 = 0

Multiply both sides

2 (r cos (θ))^{2} - cos (θ) \times r^{2} - r^{2} = 0

Use substitution

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Evaluate

2 (r cos (θ))^{2} - cos (θ) \times r^{2} - r^{2}

To covert the equation to rectangular coordinates using conversion formulas,substitute r cos θ for x

2 x^{2} - cos (θ) \times r^{2} - r^{2}

Use substitution

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Evaluate

- cos (θ) \times r^{2}

Use the commutative property to reorder the terms

r^{2} cos (θ) (- 1)

To covert the equation to rectangular coordinates using conversion formulas,substitute r cos θ for x

r x (- 1)

Evaluate

x r (- 1)

Multiply the terms

- x r

2 x^{2} - x r - r^{2}

To covert the equation to rectangular coordinates using conversion formulas,substitute x^{2} + y^{2} for r^{2}

2 x^{2} - x r - (x^{2} + y^{2})

Simplify the expression

x^{2} - x r - y^{2}

x^{2} - x r - y^{2} = 0

Simplify the expression

- x r = - x^{2} + y^{2}

Square both sides of the equation

(- x r)^{2} = (- x^{2} + y^{2})^{2}

Evaluate

(- x)^{2} r^{2} = (- x^{2} + y^{2})^{2}

To covert the equation to rectangular coordinates using conversion formulas,substitute x^{2} + y^{2} for r^{2}

(- x)^{2} (x^{2} + y^{2}) = (- x^{2} + y^{2})^{2}

Use substitution

x^{2} (x^{2} + y^{2}) = (- x^{2} + y^{2})^{2}

Calculate

x^{4} + x^{2} y^{2} = x^{4} - 2 x^{2} y^{2} + y^{4}

Move the expression to the left side

x^{4} + x^{2} y^{2} - (x^{4} - 2 x^{2} y^{2}) = y^{4}

Calculate

0 + x^{2} y^{2} = - 2 x^{2} y^{2} + y^{4}

Calculate

More Steps

Evaluate

x^{2} y^{2} + 2 x^{2} y^{2}

Collect like terms by calculating the sum or difference of their coefficients

(1 + 2) x^{2} y^{2}

Add the numbers

3 x^{2} y^{2}

0 + 3 x^{2} y^{2} = y^{4}

Solution

3 x^{2} y^{2} = y^{4}

Show Solution