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

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

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Popular Trigonometry >

3/2 sec(θ)+3cos(θ)=3

Solution

\frac{3}{2} sec (θ) + 3 cos (θ) = 3

Solution

No Solution for θ \in R

Solution steps

\frac{3}{2} sec (θ) + 3 cos (θ) = 3

Subtract

3

from both sides

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

Simplify

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

\frac{3}{2} sec (θ) + 3 cos (θ) - 3

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}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

3 \cdot 2 = 6

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

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

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

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

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

Simplify

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

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

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 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 \cdot 3}

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 3 = 6

= \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

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 3 = 6

= \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

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

Graph

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Frequently Asked Questions (FAQ)

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