What is the general solution for 3cos(x)=2sec(x)-5 ?

The general solution for 3cos(x)=2sec(x)-5 is x=1.23095…+2pin,x=2pi-1.23095…+2pin

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3cos(x)=2sec(x)-5

Solution

3 cos (x) = 2 sec (x) - 5

Solution

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Degrees

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 cos (x) = 2 sec (x) - 5

Subtract

2 sec (x) - 5

from both sides

3 cos (x) - 2 sec (x) + 5 = 0

Rewrite using trig identities

5 - 2 sec (x) + 3 cos (x)

Use the basic trigonometric identity:

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

= 5 - 2 sec (x) + 3 \cdot \frac{1}{sec ( x )}

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

3 \cdot \frac{1}{sec ( x )}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{sec ( x )}

= 5 - 2 sec (x) + \frac{3}{sec ( x )}

5 + \frac{3}{sec ( x )} - 2 sec (x) = 0

Solve by substitution

5 + \frac{3}{sec ( x )} - 2 sec (x) = 0

Let:

sec (x) = u

5 + \frac{3}{u} - 2 u = 0

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

5 + \frac{3}{u} - 2 u = 0

Multiply both sides by

u

5 + \frac{3}{u} - 2 u = 0

Multiply both sides by

u

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

Simplify

5 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

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

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

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

Solve

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

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

Write in the standard form

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

- 2 u^{2} + 5 u + 3 = 0

Solve with the quadratic formula

- 2 u^{2} + 5 u + 3 = 0

Quadratic Equation Formula:

For

a = - 2, b = 5, c = 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

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

5^{2} - 4 (- 2) \cdot 3

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

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

Separate the solutions

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

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

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

Remove parentheses:

(- a) = - a

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

Add/Subtract the numbers:

- 5 + 7 = 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{- 5 - 7}{2 ( - 2 )} : 3

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

Remove parentheses:

(- a) = - a

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

Subtract the numbers:

- 5 - 7 = - 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

5 + \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 (x)

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

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

sec (x) = - \frac{1}{2} :

No Solution

sec (x) = - \frac{1}{2}

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

No Solution

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

sec (x) = 3

Apply trig inverse properties

sec (x) = 3

General solutions for

sec (x) = 3

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

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

Combine all the solutions

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

Show solutions in decimal form

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos(x)=2sec(x)-5 ?
The general solution for 3cos(x)=2sec(x)-5 is x=1.23095…+2pin,x=2pi-1.23095…+2pin