What is the general solution for 3cos(θ)+2sec(θ)+7=0 ?

The general solution for 3cos(θ)+2sec(θ)+7=0 is θ=1.91063…+2pin,θ=-1.91063…+2pin

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Popular >

3cos(θ)+2sec(θ)+7=0

Solution

3 cos (θ) + 2 sec (θ) + 7 = 0

Solution

θ = 1.91063 \dots + 2 πn, θ = - 1.91063 \dots + 2 πn

Degrees

θ = 109.47122 \dots^{\circ} + 36 0^{\circ} n, θ = - 109.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 cos (θ) + 2 sec (θ) + 7 = 0

Rewrite using trig identities

7 + 2 sec (θ) + 3 cos (θ)

Use the basic trigonometric identity:

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

= 7 + 2 sec (θ) + 3 \cdot \frac{1}{sec ( θ )}

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

3 \cdot \frac{1}{sec ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 3}{sec ( θ )}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{sec ( θ )}

= 7 + 2 sec (θ) + \frac{3}{sec ( θ )}

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

Solve by substitution

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

Let:

sec (θ) = u

7 + \frac{3}{u} + 2 u = 0

7 + \frac{3}{u} + 2 u = 0 : u = - \frac{1}{2}, u = - 3

7 + \frac{3}{u} + 2 u = 0

Multiply both sides by

u

7 + \frac{3}{u} + 2 u = 0

Multiply both sides by

u

7 u + \frac{3}{u} u + 2 uu = 0 \cdot u

Simplify

7 u + \frac{3}{u} u + 2 uu = 0 \cdot u

Simplify

\frac{3}{u} u : 3

\frac{3}{u} u

Multiply fractions:

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

= \frac{3 u}{u}

Cancel the common factor:

u

= 3

Simplify

2 uu : 2 u^{2}

2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

7 u + 3 + 2 u^{2} = 0

7 u + 3 + 2 u^{2} = 0

7 u + 3 + 2 u^{2} = 0

Solve

7 u + 3 + 2 u^{2} = 0 : u = - \frac{1}{2}, u = - 3

7 u + 3 + 2 u^{2} = 0

Write in the standard form

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

2 u^{2} + 7 u + 3 = 0

Solve with the quadratic formula

2 u^{2} + 7 u + 3 = 0

Quadratic Equation Formula:

For

a = 2, b = 7, c = 3

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

7^{2} - 4 \cdot 2 \cdot 3 = 5

7^{2} - 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

Subtract the numbers:

49 - 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

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

Separate the solutions

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

u = \frac{- 7 + 5}{2 \cdot 2} : - \frac{1}{2}

\frac{- 7 + 5}{2 \cdot 2}

Add/Subtract the numbers:

- 7 + 5 = - 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

u = \frac{- 7 - 5}{2 \cdot 2} : - 3

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

Subtract the numbers:

- 7 - 5 = - 12

= \frac{- 12}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 12}{4}

Apply the fraction rule:

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

= - \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= - 3

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

7 + \frac{3}{u} + 2 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = sec (θ)

sec (θ) = - \frac{1}{2}, sec (θ) = - 3

sec (θ) = - \frac{1}{2}, sec (θ) = - 3

sec (θ) = - \frac{1}{2} :

No Solution

sec (θ) = - \frac{1}{2}

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

sec (θ) = - 3 : θ = arcsec (- 3) + 2 πn, θ = - arcsec (- 3) + 2 πn

sec (θ) = - 3

Apply trig inverse properties

sec (θ) = - 3

General solutions for

sec (θ) = - 3

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

θ = arcsec (- 3) + 2 πn, θ = - arcsec (- 3) + 2 πn

θ = arcsec (- 3) + 2 πn, θ = - arcsec (- 3) + 2 πn

Combine all the solutions

θ = arcsec (- 3) + 2 πn, θ = - arcsec (- 3) + 2 πn

Show solutions in decimal form

θ = 1.91063 \dots + 2 πn, θ = - 1.91063 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos(θ)+2sec(θ)+7=0 ?
The general solution for 3cos(θ)+2sec(θ)+7=0 is θ=1.91063…+2pin,θ=-1.91063…+2pin