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Popular >

7cos^2(x)-1=5cos(x),0<= x<= 2pi

Solution

7 cos^{2} (x) - 1 = 5 cos (x), 0 \leq x \leq 2 π

Solution

x = 0.50090 \dots, x = 2 π - 0.50090 \dots, x = 1.73439 \dots, x = - 1.73439 \dots + 2 π

Degrees

x = 28.69945 \dots^{\circ}, x = 331.30054 \dots^{\circ}, x = 99.37322 \dots^{\circ}, x = 260.62677 \dots^{\circ}

Solution steps

7 cos^{2} (x) - 1 = 5 cos (x), 0 \leq x \leq 2 π

Solve by substitution

7 cos^{2} (x) - 1 = 5 cos (x)

Let:

cos (x) = u

7 u^{2} - 1 = 5 u

7 u^{2} - 1 = 5 u : u = \frac{5 + 53}{14}, u = \frac{5 - 53}{14}

7 u^{2} - 1 = 5 u

Move

5 u

to the left side

7 u^{2} - 1 = 5 u

Subtract

5 u

from both sides

7 u^{2} - 1 - 5 u = 5 u - 5 u

Simplify

7 u^{2} - 1 - 5 u = 0

7 u^{2} - 1 - 5 u = 0

Write in the standard form

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

7 u^{2} - 5 u - 1 = 0

Solve with the quadratic formula

7 u^{2} - 5 u - 1 = 0

Quadratic Equation Formula:

For

a = 7, b = - 5, c = - 1

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 7 ( - 1 )}{2 \cdot 7}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 7 ( - 1 )}{2 \cdot 7}

(- 5)^{2} - 4 \cdot 7 (- 1) = 53

(- 5)^{2} - 4 \cdot 7 (- 1)

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 7 \cdot 1

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 7 \cdot 1

Multiply the numbers:

4 \cdot 7 \cdot 1 = 28

= 5^{2} + 28

5^{2} = 25

= 25 + 28

Add the numbers:

25 + 28 = 53

= 53

u_{1, 2} = \frac{- ( - 5 ) \pm 53}{2 \cdot 7}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 53}{2 \cdot 7}, u_{2} = \frac{- ( - 5 ) - 53}{2 \cdot 7}

u = \frac{- ( - 5 ) + 53}{2 \cdot 7} : \frac{5 + 53}{14}

\frac{- ( - 5 ) + 53}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{5 + 53}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{5 + 53}{14}

u = \frac{- ( - 5 ) - 53}{2 \cdot 7} : \frac{5 - 53}{14}

\frac{- ( - 5 ) - 53}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{5 - 53}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{5 - 53}{14}

The solutions to the quadratic equation are:

u = \frac{5 + 53}{14}, u = \frac{5 - 53}{14}

Substitute back

u = cos (x)

cos (x) = \frac{5 + 53}{14}, cos (x) = \frac{5 - 53}{14}

cos (x) = \frac{5 + 53}{14}, cos (x) = \frac{5 - 53}{14}

cos (x) = \frac{5 + 53}{14}, 0 \leq x \leq 2 π : x = arccos (\frac{5 + 53}{14}), x = 2 π - arccos (\frac{5 + 53}{14})

cos (x) = \frac{5 + 53}{14}, 0 \leq x \leq 2 π

Apply trig inverse properties

cos (x) = \frac{5 + 53}{14}

General solutions for

cos (x) = \frac{5 + 53}{14}

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

x = arccos (\frac{5 + 53}{14}) + 2 πn, x = 2 π - arccos (\frac{5 + 53}{14}) + 2 πn

x = arccos (\frac{5 + 53}{14}) + 2 πn, x = 2 π - arccos (\frac{5 + 53}{14}) + 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = arccos (\frac{5 + 53}{14}), x = 2 π - arccos (\frac{5 + 53}{14})

cos (x) = \frac{5 - 53}{14}, 0 \leq x \leq 2 π : x = arccos (\frac{5 - 53}{14}), x = - arccos (\frac{5 - 53}{14}) + 2 π

cos (x) = \frac{5 - 53}{14}, 0 \leq x \leq 2 π

Apply trig inverse properties

cos (x) = \frac{5 - 53}{14}

General solutions for

cos (x) = \frac{5 - 53}{14}

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

x = arccos (\frac{5 - 53}{14}) + 2 πn, x = - arccos (\frac{5 - 53}{14}) + 2 πn

x = arccos (\frac{5 - 53}{14}) + 2 πn, x = - arccos (\frac{5 - 53}{14}) + 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = arccos (\frac{5 - 53}{14}), x = - arccos (\frac{5 - 53}{14}) + 2 π

Combine all the solutions

x = arccos (\frac{5 + 53}{14}), x = 2 π - arccos (\frac{5 + 53}{14}), x = arccos (\frac{5 - 53}{14}), x = - arccos (\frac{5 - 53}{14}) + 2 π

Show solutions in decimal form

x = 0.50090 \dots, x = 2 π - 0.50090 \dots, x = 1.73439 \dots, x = - 1.73439 \dots + 2 π

Graph

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