What is the general solution for 229cos(a)+14715sin(a)=3959.55 ?

The general solution for 229cos(a)+14715sin(a)=3959.55 is a=0.25684…+2pin,a=pi-0.28796…+2pin

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Popular >

229cos(a)+14715sin(a)=3959.55

Solution

229 cos (a) + 14715 sin (a) = 3959.55

Solution

a = 0.25684 \dots + 2 πn, a = π - 0.28796 \dots + 2 πn

Degrees

a = 14.71615 \dots^{\circ} + 36 0^{\circ} n, a = 163.50067 \dots^{\circ} + 36 0^{\circ} n

Solution steps

229 cos (a) + 14715 sin (a) = 3959.55

Subtract

14715 sin (a)

from both sides

229 cos (a) = 3959.55 - 14715 sin (a)

Square both sides

(229 cos (a))^{2} = (3959.55 - 14715 sin (a))^{2}

Subtract

(3959.55 - 14715 sin (a))^{2}

from both sides

52441 cos^{2} (a) - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) = 0

Rewrite using trig identities

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 cos^{2} (a)

Use the Pythagorean identity:

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

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

= - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a))

Simplify

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a)) : 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a))

Expand

52441 (1 - sin^{2} (a)) : 52441 - 52441 sin^{2} (a)

52441 (1 - sin^{2} (a))

Apply the distributive law:

a (b - c) = ab - a c

a = 52441, b = 1, c = sin^{2} (a)

= 52441 \cdot 1 - 52441 sin^{2} (a)

Multiply the numbers:

52441 \cdot 1 = 52441

= 52441 - 52441 sin^{2} (a)

= - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a)

Simplify

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a) : 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a)

Group like terms

= 116529556.5 sin (a) - 216531225 sin^{2} (a) - 52441 sin^{2} (a) - 15678036.2025 + 52441

Add similar elements:

- 216531225 sin^{2} (a) - 52441 sin^{2} (a) = - 216583666 sin^{2} (a)

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15678036.2025 + 52441

Add/Subtract the numbers:

- 15678036.2025 + 52441 = - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15625595.2025 + 116529556.5 sin (a) - 216583666 sin^{2} (a) = 0

Solve by substitution

- 15625595.2025 + 116529556.5 sin (a) - 216583666 sin^{2} (a) = 0

Let:

sin (a) = u

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0 : u = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, u = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0

Write in the standard form

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

- 216583666 u^{2} + 116529556.5 u - 15625595.2025 = 0

Solve with the quadratic formula

- 216583666 u^{2} + 116529556.5 u - 15625595.2025 = 0

Quadratic Equation Formula:

For

a = - 216583666, b = 116529556.5, c = - 15625595.2025

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 4 ( - 216583666 ) ( - 15625595.2025 )}{2 ( - 216583666 )}

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 4 ( - 216583666 ) ( - 15625595.2025 )}{2 ( - 216583666 )}

116529556. 5^{2} - 4 (- 216583666) (- 15625595.2025) = 116529556. 5^{2} - 1.3537 E 16

116529556. 5^{2} - 4 (- 216583666) (- 15625595.2025)

Apply rule

- (- a) = a

= 116529556. 5^{2} - 4 \cdot 216583666 \cdot 15625595.2025

Multiply the numbers:

4 \cdot 216583666 \cdot 15625595.2025 = 1.3537 E 16

= 116529556. 5^{2} - 1.3537 E 16

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

Separate the solutions

u_{1} = \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}, u_{2} = \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

u = \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )} : \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

\frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

Remove parentheses:

(- a) = - a

= \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{- 2 \cdot 216583666}

Multiply the numbers:

2 \cdot 216583666 = 433167332

= \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{- 433167332}

Apply the fraction rule:

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

- 116529556.5 + 116529556. 5^{2} - 1.3537 E 16 = - (- 116529556. 5^{2} - 1.3537 E 16 + 116529556.5)

= \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

u = \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )} : \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

\frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

Remove parentheses:

(- a) = - a

= \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{- 2 \cdot 216583666}

Multiply the numbers:

2 \cdot 216583666 = 433167332

= \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{- 433167332}

Apply the fraction rule:

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

- 116529556.5 - 116529556. 5^{2} - 1.3537 E 16 = - (116529556. 5^{2} - 1.3537 E 16 + 116529556.5)

= \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

The solutions to the quadratic equation are:

u = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, u = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Substitute back

u = sin (a)

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332} : a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Apply trig inverse properties

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

General solutions for

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

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

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332} : a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Apply trig inverse properties

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

General solutions for

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

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

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Combine all the solutions

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

229 cos (a) + 14715 sin (a) = 3959.55

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

Check the solution

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn :

True

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Plug in

n = 1

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

For

229 cos (a) + 14715 sin (a) = 3959.55

plug in

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

Refine

3959.54999 \dots = 3959.55

\Rightarrow True

Check the solution

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn :

False

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

For

229 cos (a) + 14715 sin (a) = 3959.55

plug in

a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

Refine

3516.57416 \dots = 3959.55

\Rightarrow False

Check the solution

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn :

False

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Plug in

n = 1

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

For

229 cos (a) + 14715 sin (a) = 3959.55

plug in

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

Refine

4398.69096 \dots = 3959.55

\Rightarrow False

Check the solution

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn :

True

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

For

229 cos (a) + 14715 sin (a) = 3959.55

plug in

a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

Refine

3959.54999 \dots = 3959.55

\Rightarrow True

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Show solutions in decimal form

a = 0.25684 \dots + 2 πn, a = π - 0.28796 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 229cos(a)+14715sin(a)=3959.55 ?
The general solution for 229cos(a)+14715sin(a)=3959.55 is a=0.25684…+2pin,a=pi-0.28796…+2pin