What is the general solution for 5sin(x)-12cos(x)=13 ?

The general solution for 5sin(x)-12cos(x)=13 is x=2.74680…+2pin

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Popular Trigonometry >

5sin(x)-12cos(x)=13

Solution

5 sin (x) - 12 cos (x) = 13

Solution

x = 2.74680 \dots + 2 πn

Degrees

x = 157.38013 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 sin (x) - 12 cos (x) = 13

Add

12 cos (x)

to both sides

5 sin (x) = 13 + 12 cos (x)

Square both sides

(5 sin (x))^{2} = (13 + 12 cos (x))^{2}

Subtract

(13 + 12 cos (x))^{2}

from both sides

25 sin^{2} (x) - 169 - 312 cos (x) - 144 cos^{2} (x) = 0

Rewrite using trig identities

- 169 - 144 cos^{2} (x) + 25 sin^{2} (x) - 312 cos (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 169 - 144 cos^{2} (x) + 25 (1 - cos^{2} (x)) - 312 cos (x)

Simplify

- 169 - 144 cos^{2} (x) + 25 (1 - cos^{2} (x)) - 312 cos (x) : - 169 cos^{2} (x) - 312 cos (x) - 144

- 169 - 144 cos^{2} (x) + 25 (1 - cos^{2} (x)) - 312 cos (x)

Expand

25 (1 - cos^{2} (x)) : 25 - 25 cos^{2} (x)

25 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 25, b = 1, c = cos^{2} (x)

= 25 \cdot 1 - 25 cos^{2} (x)

Multiply the numbers:

25 \cdot 1 = 25

= 25 - 25 cos^{2} (x)

= - 169 - 144 cos^{2} (x) + 25 - 25 cos^{2} (x) - 312 cos (x)

Simplify

- 169 - 144 cos^{2} (x) + 25 - 25 cos^{2} (x) - 312 cos (x) : - 169 cos^{2} (x) - 312 cos (x) - 144

- 169 - 144 cos^{2} (x) + 25 - 25 cos^{2} (x) - 312 cos (x)

Group like terms

= - 144 cos^{2} (x) - 25 cos^{2} (x) - 312 cos (x) - 169 + 25

Add similar elements:

- 144 cos^{2} (x) - 25 cos^{2} (x) = - 169 cos^{2} (x)

= - 169 cos^{2} (x) - 312 cos (x) - 169 + 25

Add/Subtract the numbers:

- 169 + 25 = - 144

= - 169 cos^{2} (x) - 312 cos (x) - 144

= - 169 cos^{2} (x) - 312 cos (x) - 144

= - 169 cos^{2} (x) - 312 cos (x) - 144

- 144 - 169 cos^{2} (x) - 312 cos (x) = 0

Solve by substitution

- 144 - 169 cos^{2} (x) - 312 cos (x) = 0

Let:

cos (x) = u

- 144 - 169 u^{2} - 312 u = 0

- 144 - 169 u^{2} - 312 u = 0 : u = - \frac{12}{13}

- 144 - 169 u^{2} - 312 u = 0

Write in the standard form

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

- 169 u^{2} - 312 u - 144 = 0

Solve with the quadratic formula

- 169 u^{2} - 312 u - 144 = 0

Quadratic Equation Formula:

For

a = - 169, b = - 312, c = - 144

u_{1, 2} = \frac{- ( - 312 ) \pm ( - 312 ) ^{2} - 4 ( - 169 ) ( - 144 )}{2 ( - 169 )}

u_{1, 2} = \frac{- ( - 312 ) \pm ( - 312 ) ^{2} - 4 ( - 169 ) ( - 144 )}{2 ( - 169 )}

(- 312)^{2} - 4 (- 169) (- 144) = 0

(- 312)^{2} - 4 (- 169) (- 144)

Apply rule

- (- a) = a

= (- 312)^{2} - 4 \cdot 169 \cdot 144

Apply exponent rule:

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

n

is even

(- 312)^{2} = 31 2^{2}

= 31 2^{2} - 4 \cdot 169 \cdot 144

Multiply the numbers:

4 \cdot 169 \cdot 144 = 97344

= 31 2^{2} - 97344

31 2^{2} = 97344

= 97344 - 97344

Subtract the numbers:

97344 - 97344 = 0

= 0

u_{1, 2} = \frac{- ( - 312 ) \pm 0}{2 ( - 169 )}

u = \frac{- ( - 312 )}{2 ( - 169 )}

\frac{- ( - 312 )}{2 ( - 169 )} = - \frac{12}{13}

\frac{- ( - 312 )}{2 ( - 169 )}

Remove parentheses:

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

= \frac{312}{- 2 \cdot 169}

Multiply the numbers:

2 \cdot 169 = 338

= \frac{312}{- 338}

Apply the fraction rule:

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

= - \frac{312}{338}

Cancel the common factor:

26

= - \frac{12}{13}

u = - \frac{12}{13}

The solution to the quadratic equation is:

u = - \frac{12}{13}

Substitute back

u = cos (x)

cos (x) = - \frac{12}{13}

cos (x) = - \frac{12}{13}

cos (x) = - \frac{12}{13} : x = arccos (- \frac{12}{13}) + 2 πn, x = - arccos (- \frac{12}{13}) + 2 πn

cos (x) = - \frac{12}{13}

Apply trig inverse properties

cos (x) = - \frac{12}{13}

General solutions for

cos (x) = - \frac{12}{13}

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

x = arccos (- \frac{12}{13}) + 2 πn, x = - arccos (- \frac{12}{13}) + 2 πn

x = arccos (- \frac{12}{13}) + 2 πn, x = - arccos (- \frac{12}{13}) + 2 πn

Combine all the solutions

x = arccos (- \frac{12}{13}) + 2 πn, x = - arccos (- \frac{12}{13}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

5 sin (x) - 12 cos (x) = 13

Remove the ones that don't agree with the equation.

Check the solution

arccos (- \frac{12}{13}) + 2 πn :

True

arccos (- \frac{12}{13}) + 2 πn

Plug in

n = 1

arccos (- \frac{12}{13}) + 2 π 1

For

5 sin (x) - 12 cos (x) = 13

plug in

x = arccos (- \frac{12}{13}) + 2 π 1

5 sin (arccos (- \frac{12}{13}) + 2 π 1) - 12 cos (arccos (- \frac{12}{13}) + 2 π 1) = 13

Refine

13 = 13

\Rightarrow True

Check the solution

- arccos (- \frac{12}{13}) + 2 πn :

False

- arccos (- \frac{12}{13}) + 2 πn

Plug in

n = 1

- arccos (- \frac{12}{13}) + 2 π 1

For

5 sin (x) - 12 cos (x) = 13

plug in

x = - arccos (- \frac{12}{13}) + 2 π 1

5 sin (- arccos (- \frac{12}{13}) + 2 π 1) - 12 cos (- arccos (- \frac{12}{13}) + 2 π 1) = 13

Refine

9.15384 \dots = 13

\Rightarrow False

x = arccos (- \frac{12}{13}) + 2 πn

Show solutions in decimal form

x = 2.74680 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 5sin(x)-12cos(x)=13 ?
The general solution for 5sin(x)-12cos(x)=13 is x=2.74680…+2pin