What is the general solution for 3-3cos(θ)= 3/2 sec(θ) ?

The general solution for 3-3cos(θ)= 3/2 sec(θ) is No Solution for θ\in\mathbb{R}

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Popular >

3-3cos(θ)= 3/2 sec(θ)

Solution

3 - 3 cos (θ) = \frac{3}{2} sec (θ)

Solution

No Solution for θ \in R

Solution steps

3 - 3 cos (θ) = \frac{3}{2} sec (θ)

Subtract

\frac{3}{2} sec (θ)

from both sides

3 - 3 cos (θ) - \frac{3}{2} sec (θ) = 0

Simplify

3 - 3 cos (θ) - \frac{3}{2} sec (θ) : \frac{6 - 6 cos ( θ ) - 3 sec ( θ )}{2}

3 - 3 cos (θ) - \frac{3}{2} sec (θ)

Multiply

\frac{3}{2} sec (θ) : \frac{3 sec ( θ )}{2}

\frac{3}{2} sec (θ)

Multiply fractions:

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

= \frac{3 sec ( θ )}{2}

= 3 - 3 cos (θ) - \frac{3 sec ( θ )}{2}

Convert element to fraction:

3 = \frac{3 \cdot 2}{2}, 3 cos (θ) = \frac{3 cos ( θ ) 2}{2}

= \frac{3 \cdot 2}{2} - \frac{3 cos ( θ ) \cdot 2}{2} - \frac{3 sec ( θ )}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{3 \cdot 2 - 3 cos ( θ ) \cdot 2 - 3 sec ( θ )}{2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 - 6 cos ( θ ) - 3 sec ( θ )}{2}

\frac{6 - 6 cos ( θ ) - 3 sec ( θ )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

6 - 6 cos (θ) - 3 sec (θ) = 0

Rewrite using trig identities

6 - 3 sec (θ) - 6 cos (θ)

Use the basic trigonometric identity:

cos (x) = \frac{1}{sec ( x )}

= 6 - 3 sec (θ) - 6 \cdot \frac{1}{sec ( θ )}

6 \cdot \frac{1}{sec ( θ )} = \frac{6}{sec ( θ )}

6 \cdot \frac{1}{sec ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 6}{sec ( θ )}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{sec ( θ )}

= 6 - 3 sec (θ) - \frac{6}{sec ( θ )}

6 - \frac{6}{sec ( θ )} - 3 sec (θ) = 0

Solve by substitution

6 - \frac{6}{sec ( θ )} - 3 sec (θ) = 0

Let:

sec (θ) = u

6 - \frac{6}{u} - 3 u = 0

6 - \frac{6}{u} - 3 u = 0 : u = 1 - i, u = 1 + i

6 - \frac{6}{u} - 3 u = 0

Multiply both sides by

u

6 - \frac{6}{u} - 3 u = 0

Multiply both sides by

u

6 u - \frac{6}{u} u - 3 uu = 0 \cdot u

Simplify

6 u - \frac{6}{u} u - 3 uu = 0 \cdot u

Simplify

- \frac{6}{u} u : - 6

- \frac{6}{u} u

Multiply fractions:

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

= - \frac{6 u}{u}

Cancel the common factor:

u

= - 6

Simplify

- 3 uu : - 3 u^{2}

- 3 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= - 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

6 u - 6 - 3 u^{2} = 0

6 u - 6 - 3 u^{2} = 0

6 u - 6 - 3 u^{2} = 0

Solve

6 u - 6 - 3 u^{2} = 0 : u = 1 - i, u = 1 + i

6 u - 6 - 3 u^{2} = 0

Write in the standard form

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

- 3 u^{2} + 6 u - 6 = 0

Solve with the quadratic formula

- 3 u^{2} + 6 u - 6 = 0

Quadratic Equation Formula:

For

a = - 3, b = 6, c = - 6

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 3 ) ( - 6 )}{2 ( - 3 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 3 ) ( - 6 )}{2 ( - 3 )}

Simplify

6^{2} - 4 (- 3) (- 6) : 6 i

6^{2} - 4 (- 3) (- 6)

Apply rule

- (- a) = a

= 6^{2} - 4 \cdot 3 \cdot 6

Multiply the numbers:

4 \cdot 3 \cdot 6 = 72

= 6^{2} - 72

Apply imaginary number rule:

- a = i a

= i 72 - 6^{2}

- 6^{2} + 72 = 6

- 6^{2} + 72

6^{2} = 36

= - 36 + 72

Add/Subtract the numbers:

- 36 + 72 = 36

= 36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

= 6 i

u_{1, 2} = \frac{- 6 \pm 6 i}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- 6 + 6 i}{2 ( - 3 )}, u_{2} = \frac{- 6 - 6 i}{2 ( - 3 )}

u = \frac{- 6 + 6 i}{2 ( - 3 )} : 1 - i

\frac{- 6 + 6 i}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 + 6 i}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 6 + 6 i}{- 6}

Apply the fraction rule:

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

= - \frac{- 6 + 6 i}{6}

Cancel

\frac{- 6 + 6 i}{6} : - 1 + i

\frac{- 6 + 6 i}{6}

Factor

- 6 + 6 i : 6 (- 1 + i)

- 6 + 6 i

Rewrite as

= - 6 \cdot 1 + 6 i

Factor out common term

6

= 6 (- 1 + i)

= \frac{6 ( - 1 + i )}{6}

Divide the numbers:

\frac{6}{6} = 1

= - 1 + i

= - (- 1 + i)

Distribute parentheses

= - (- 1) - (i)

Apply minus-plus rules

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

= 1 - i

u = \frac{- 6 - 6 i}{2 ( - 3 )} : 1 + i

\frac{- 6 - 6 i}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 - 6 i}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 6 - 6 i}{- 6}

Apply the fraction rule:

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

= - \frac{- 6 - 6 i}{6}

Cancel

\frac{- 6 - 6 i}{6} : - 1 - i

\frac{- 6 - 6 i}{6}

Factor

- 6 - 6 i : - 6 (1 + i)

- 6 - 6 i

Rewrite as

= - 6 \cdot 1 - 6 i

Factor out common term

6

= - 6 (1 + i)

= - \frac{6 ( 1 + i )}{6}

Divide the numbers:

\frac{6}{6} = 1

= - (1 + i)

Negate

- (1 + i) = - 1 - i

= - 1 - i

= - (- 1 - i)

Distribute parentheses

= - (- 1) - (- i)

Apply minus-plus rules

- (- a) = a

= 1 + i

The solutions to the quadratic equation are:

u = 1 - i, u = 1 + i

u = 1 - i, u = 1 + i

Substitute back

u = sec (θ)

sec (θ) = 1 - i, sec (θ) = 1 + i

sec (θ) = 1 - i, sec (θ) = 1 + i

sec (θ) = 1 - i :

No Solution

sec (θ) = 1 - i

No Solution

sec (θ) = 1 + i :

No Solution

sec (θ) = 1 + i

No Solution

Combine all the solutions

No Solution for θ \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 3-3cos(θ)= 3/2 sec(θ) ?
The general solution for 3-3cos(θ)= 3/2 sec(θ) is No Solution for θ\in\mathbb{R}