What is the general solution for 12cos(β)-5sin(β)=4.7 ?

The general solution for 12cos(β)-5sin(β)=4.7 is β=pi+1.54592…+2pin,β=0.80608…+2pin

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Popular Trigonometry >

12cos(β)-5sin(β)=4.7

Solution

12 cos (β) - 5 sin (β) = 4.7

Solution

β = π + 1.54592 \dots + 2 πn, β = 0.80608 \dots + 2 πn

Degrees

β = 268.57484 \dots^{\circ} + 36 0^{\circ} n, β = 46.18542 \dots^{\circ} + 36 0^{\circ} n

Solution steps

12 cos (β) - 5 sin (β) = 4.7

Add

5 sin (β)

to both sides

12 cos (β) = 4.7 + 5 sin (β)

Square both sides

(12 cos (β))^{2} = (4.7 + 5 sin (β))^{2}

Subtract

(4.7 + 5 sin (β))^{2}

from both sides

144 cos^{2} (β) - 22.09 - 47 sin (β) - 25 sin^{2} (β) = 0

Rewrite using trig identities

- 22.09 + 144 cos^{2} (β) - 25 sin^{2} (β) - 47 sin (β)

Use the Pythagorean identity:

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

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

= - 22.09 + 144 (1 - sin^{2} (β)) - 25 sin^{2} (β) - 47 sin (β)

Simplify

- 22.09 + 144 (1 - sin^{2} (β)) - 25 sin^{2} (β) - 47 sin (β) : - 169 sin^{2} (β) - 47 sin (β) + 121.91

- 22.09 + 144 (1 - sin^{2} (β)) - 25 sin^{2} (β) - 47 sin (β)

Expand

144 (1 - sin^{2} (β)) : 144 - 144 sin^{2} (β)

144 (1 - sin^{2} (β))

Apply the distributive law:

a (b - c) = ab - a c

a = 144, b = 1, c = sin^{2} (β)

= 144 \cdot 1 - 144 sin^{2} (β)

Multiply the numbers:

144 \cdot 1 = 144

= 144 - 144 sin^{2} (β)

= - 22.09 + 144 - 144 sin^{2} (β) - 25 sin^{2} (β) - 47 sin (β)

Simplify

- 22.09 + 144 - 144 sin^{2} (β) - 25 sin^{2} (β) - 47 sin (β) : - 169 sin^{2} (β) - 47 sin (β) + 121.91

- 22.09 + 144 - 144 sin^{2} (β) - 25 sin^{2} (β) - 47 sin (β)

Add similar elements:

- 144 sin^{2} (β) - 25 sin^{2} (β) = - 169 sin^{2} (β)

= - 22.09 + 144 - 169 sin^{2} (β) - 47 sin (β)

Add/Subtract the numbers:

- 22.09 + 144 = 121.91

= - 169 sin^{2} (β) - 47 sin (β) + 121.91

= - 169 sin^{2} (β) - 47 sin (β) + 121.91

= - 169 sin^{2} (β) - 47 sin (β) + 121.91

121.91 - 169 sin^{2} (β) - 47 sin (β) = 0

Solve by substitution

121.91 - 169 sin^{2} (β) - 47 sin (β) = 0

Let:

sin (β) = u

121.91 - 169 u^{2} - 47 u = 0

121.91 - 169 u^{2} - 47 u = 0 : u = - \frac{4700 + 846201600}{33800}, u = \frac{846201600 - 4700}{33800}

121.91 - 169 u^{2} - 47 u = 0

Multiply both sides by

100

121.91 - 169 u^{2} - 47 u = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

121.91 \cdot 100 - 169 u^{2} \cdot 100 - 47 u \cdot 100 = 0 \cdot 100

Refine

12191 - 16900 u^{2} - 4700 u = 0

12191 - 16900 u^{2} - 4700 u = 0

Write in the standard form

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

- 16900 u^{2} - 4700 u + 12191 = 0

Solve with the quadratic formula

- 16900 u^{2} - 4700 u + 12191 = 0

Quadratic Equation Formula:

For

a = - 16900, b = - 4700, c = 12191

u_{1, 2} = \frac{- ( - 4700 ) \pm ( - 4700 ) ^{2} - 4 ( - 16900 ) \cdot 12191}{2 ( - 16900 )}

u_{1, 2} = \frac{- ( - 4700 ) \pm ( - 4700 ) ^{2} - 4 ( - 16900 ) \cdot 12191}{2 ( - 16900 )}

(- 4700)^{2} - 4 (- 16900) \cdot 12191 = 846201600

(- 4700)^{2} - 4 (- 16900) \cdot 12191

Apply rule

- (- a) = a

= (- 4700)^{2} + 4 \cdot 16900 \cdot 12191

Apply exponent rule:

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

n

is even

(- 4700)^{2} = 470 0^{2}

= 470 0^{2} + 4 \cdot 16900 \cdot 12191

Multiply the numbers:

4 \cdot 16900 \cdot 12191 = 824111600

= 470 0^{2} + 824111600

470 0^{2} = 22090000

= 22090000 + 824111600

Add the numbers:

22090000 + 824111600 = 846201600

= 846201600

u_{1, 2} = \frac{- ( - 4700 ) \pm 846201600}{2 ( - 16900 )}

Separate the solutions

u_{1} = \frac{- ( - 4700 ) + 846201600}{2 ( - 16900 )}, u_{2} = \frac{- ( - 4700 ) - 846201600}{2 ( - 16900 )}

u = \frac{- ( - 4700 ) + 846201600}{2 ( - 16900 )} : - \frac{4700 + 846201600}{33800}

\frac{- ( - 4700 ) + 846201600}{2 ( - 16900 )}

Remove parentheses:

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

= \frac{4700 + 846201600}{- 2 \cdot 16900}

Multiply the numbers:

2 \cdot 16900 = 33800

= \frac{4700 + 846201600}{- 33800}

Apply the fraction rule:

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

= - \frac{4700 + 846201600}{33800}

u = \frac{- ( - 4700 ) - 846201600}{2 ( - 16900 )} : \frac{846201600 - 4700}{33800}

\frac{- ( - 4700 ) - 846201600}{2 ( - 16900 )}

Remove parentheses:

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

= \frac{4700 - 846201600}{- 2 \cdot 16900}

Multiply the numbers:

2 \cdot 16900 = 33800

= \frac{4700 - 846201600}{- 33800}

Apply the fraction rule:

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

4700 - 846201600 = - (846201600 - 4700)

= \frac{846201600 - 4700}{33800}

The solutions to the quadratic equation are:

u = - \frac{4700 + 846201600}{33800}, u = \frac{846201600 - 4700}{33800}

Substitute back

u = sin (β)

sin (β) = - \frac{4700 + 846201600}{33800}, sin (β) = \frac{846201600 - 4700}{33800}

sin (β) = - \frac{4700 + 846201600}{33800}, sin (β) = \frac{846201600 - 4700}{33800}

sin (β) = - \frac{4700 + 846201600}{33800} : β = arcsin (- \frac{4700 + 846201600}{33800}) + 2 πn, β = π + arcsin (\frac{4700 + 846201600}{33800}) + 2 πn

sin (β) = - \frac{4700 + 846201600}{33800}

Apply trig inverse properties

sin (β) = - \frac{4700 + 846201600}{33800}

General solutions for

sin (β) = - \frac{4700 + 846201600}{33800}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

β = arcsin (- \frac{4700 + 846201600}{33800}) + 2 πn, β = π + arcsin (\frac{4700 + 846201600}{33800}) + 2 πn

β = arcsin (- \frac{4700 + 846201600}{33800}) + 2 πn, β = π + arcsin (\frac{4700 + 846201600}{33800}) + 2 πn

sin (β) = \frac{846201600 - 4700}{33800} : β = arcsin (\frac{846201600 - 4700}{33800}) + 2 πn, β = π - arcsin (\frac{846201600 - 4700}{33800}) + 2 πn

sin (β) = \frac{846201600 - 4700}{33800}

Apply trig inverse properties

sin (β) = \frac{846201600 - 4700}{33800}

General solutions for

sin (β) = \frac{846201600 - 4700}{33800}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

β = arcsin (\frac{846201600 - 4700}{33800}) + 2 πn, β = π - arcsin (\frac{846201600 - 4700}{33800}) + 2 πn

β = arcsin (\frac{846201600 - 4700}{33800}) + 2 πn, β = π - arcsin (\frac{846201600 - 4700}{33800}) + 2 πn

Combine all the solutions

β = arcsin (- \frac{4700 + 846201600}{33800}) + 2 πn, β = π + arcsin (\frac{4700 + 846201600}{33800}) + 2 πn, β = arcsin (\frac{846201600 - 4700}{33800}) + 2 πn, β = π - arcsin (\frac{846201600 - 4700}{33800}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

12 cos (β) - 5 sin (β) = 4.7

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

Check the solution

arcsin (- \frac{4700 + 846201600}{33800}) + 2 πn :

False

arcsin (- \frac{4700 + 846201600}{33800}) + 2 πn

Plug in

n = 1

arcsin (- \frac{4700 + 846201600}{33800}) + 2 π 1

For

12 cos (β) - 5 sin (β) = 4.7

plug in

β = arcsin (- \frac{4700 + 846201600}{33800}) + 2 π 1

12 cos (arcsin (- \frac{4700 + 846201600}{33800}) + 2 π 1) - 5 sin (arcsin (- \frac{4700 + 846201600}{33800}) + 2 π 1) = 4.7

Refine

5.29690 \dots = 4.7

\Rightarrow False

Check the solution

π + arcsin (\frac{4700 + 846201600}{33800}) + 2 πn :

True

π + arcsin (\frac{4700 + 846201600}{33800}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{4700 + 846201600}{33800}) + 2 π 1

For

12 cos (β) - 5 sin (β) = 4.7

plug in

β = π + arcsin (\frac{4700 + 846201600}{33800}) + 2 π 1

12 cos (π + arcsin (\frac{4700 + 846201600}{33800}) + 2 π 1) - 5 sin (π + arcsin (\frac{4700 + 846201600}{33800}) + 2 π 1) = 4.7

Refine

4.7 = 4.7

\Rightarrow True

Check the solution

arcsin (\frac{846201600 - 4700}{33800}) + 2 πn :

True

arcsin (\frac{846201600 - 4700}{33800}) + 2 πn

Plug in

n = 1

arcsin (\frac{846201600 - 4700}{33800}) + 2 π 1

For

12 cos (β) - 5 sin (β) = 4.7

plug in

β = arcsin (\frac{846201600 - 4700}{33800}) + 2 π 1

12 cos (arcsin (\frac{846201600 - 4700}{33800}) + 2 π 1) - 5 sin (arcsin (\frac{846201600 - 4700}{33800}) + 2 π 1) = 4.7

Refine

4.7 = 4.7

\Rightarrow True

Check the solution

π - arcsin (\frac{846201600 - 4700}{33800}) + 2 πn :

False

π - arcsin (\frac{846201600 - 4700}{33800}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{846201600 - 4700}{33800}) + 2 π 1

For

12 cos (β) - 5 sin (β) = 4.7

plug in

β = π - arcsin (\frac{846201600 - 4700}{33800}) + 2 π 1

12 cos (π - arcsin (\frac{846201600 - 4700}{33800}) + 2 π 1) - 5 sin (π - arcsin (\frac{846201600 - 4700}{33800}) + 2 π 1) = 4.7

Refine

- 11.91584 \dots = 4.7

\Rightarrow False

β = π + arcsin (\frac{4700 + 846201600}{33800}) + 2 πn, β = arcsin (\frac{846201600 - 4700}{33800}) + 2 πn

Show solutions in decimal form

β = π + 1.54592 \dots + 2 πn, β = 0.80608 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 12cos(β)-5sin(β)=4.7 ?
The general solution for 12cos(β)-5sin(β)=4.7 is β=pi+1.54592…+2pin,β=0.80608…+2pin