What is the general solution for 7.9615=59.2153cos(x)-12sin(x) ?

The general solution for 7.9615=59.2153cos(x)-12sin(x) is x=pi+1.50300…+2pin,x=1.23869…+2pin

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Popular >

7.9615=59.2153cos(x)-12sin(x)

Solution

7.9615 = 59.2153 cos (x) - 12 sin (x)

Solution

x = π + 1.50300 \dots + 2 πn, x = 1.23869 \dots + 2 πn

Degrees

x = 266.11610 \dots^{\circ} + 36 0^{\circ} n, x = 70.97215 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7.9615 = 59.2153 cos (x) - 12 sin (x)

Add

12 sin (x)

to both sides

59.2153 cos (x) = 7.9615 + 12 sin (x)

Square both sides

(59.2153 cos (x))^{2} = (7.9615 + 12 sin (x))^{2}

Subtract

(7.9615 + 12 sin (x))^{2}

from both sides

3506.45175 \dots cos^{2} (x) - 63.38548 \dots - 191.076 sin (x) - 144 sin^{2} (x) = 0

Rewrite using trig identities

- 63.38548 \dots - 144 sin^{2} (x) - 191.076 sin (x) + 3506.45175 \dots cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 63.38548 \dots - 144 sin^{2} (x) - 191.076 sin (x) + 3506.45175 \dots (1 - sin^{2} (x))

Simplify

- 63.38548 \dots - 144 sin^{2} (x) - 191.076 sin (x) + 3506.45175 \dots (1 - sin^{2} (x)) : - 3650.45175 \dots sin^{2} (x) - 191.076 sin (x) + 3443.06627 \dots

- 63.38548 \dots - 144 sin^{2} (x) - 191.076 sin (x) + 3506.45175 \dots (1 - sin^{2} (x))

Expand

3506.45175 \dots (1 - sin^{2} (x)) : 3506.45175 \dots - 3506.45175 \dots sin^{2} (x)

3506.45175 \dots (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 3506.45175 \dots, b = 1, c = sin^{2} (x)

= 3506.45175 \dots \cdot 1 - 3506.45175 \dots sin^{2} (x)

= 1 \cdot 3506.45175 \dots - 3506.45175 \dots sin^{2} (x)

Multiply the numbers:

1 \cdot 3506.45175 \dots = 3506.45175 \dots

= 3506.45175 \dots - 3506.45175 \dots sin^{2} (x)

= - 63.38548 \dots - 144 sin^{2} (x) - 191.076 sin (x) + 3506.45175 \dots - 3506.45175 \dots sin^{2} (x)

Simplify

- 63.38548 \dots - 144 sin^{2} (x) - 191.076 sin (x) + 3506.45175 \dots - 3506.45175 \dots sin^{2} (x) : - 3650.45175 \dots sin^{2} (x) - 191.076 sin (x) + 3443.06627 \dots

- 63.38548 \dots - 144 sin^{2} (x) - 191.076 sin (x) + 3506.45175 \dots - 3506.45175 \dots sin^{2} (x)

Group like terms

= - 144 sin^{2} (x) - 191.076 sin (x) - 3506.45175 \dots sin^{2} (x) - 63.38548 \dots + 3506.45175 \dots

Add similar elements:

- 144 sin^{2} (x) - 3506.45175 \dots sin^{2} (x) = - 3650.45175 \dots sin^{2} (x)

= - 3650.45175 \dots sin^{2} (x) - 191.076 sin (x) - 63.38548 \dots + 3506.45175 \dots

Add/Subtract the numbers:

- 63.38548 \dots + 3506.45175 \dots = 3443.06627 \dots

= - 3650.45175 \dots sin^{2} (x) - 191.076 sin (x) + 3443.06627 \dots

= - 3650.45175 \dots sin^{2} (x) - 191.076 sin (x) + 3443.06627 \dots

= - 3650.45175 \dots sin^{2} (x) - 191.076 sin (x) + 3443.06627 \dots

3443.06627 \dots - 191.076 sin (x) - 3650.45175 \dots sin^{2} (x) = 0

Solve by substitution

3443.06627 \dots - 191.076 sin (x) - 3650.45175 \dots sin^{2} (x) = 0

Let:

sin (x) = u

3443.06627 \dots - 191.076 u - 3650.45175 \dots u^{2} = 0

3443.06627 \dots - 191.076 u - 3650.45175 \dots u^{2} = 0 : u = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}, u = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

3443.06627 \dots - 191.076 u - 3650.45175 \dots u^{2} = 0

Write in the standard form

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

- 3650.45175 \dots u^{2} - 191.076 u + 3443.06627 \dots = 0

Solve with the quadratic formula

- 3650.45175 \dots u^{2} - 191.076 u + 3443.06627 \dots = 0

Quadratic Equation Formula:

For

a = - 3650.45175 \dots, b = - 191.076, c = 3443.06627 \dots

u_{1, 2} = \frac{- ( - 191.076 ) \pm ( - 191.076 ) ^{2} - 4 ( - 3650.45175 \dots ) \cdot 3443.06627 \dots}{2 ( - 3650.45175 \dots )}

u_{1, 2} = \frac{- ( - 191.076 ) \pm ( - 191.076 ) ^{2} - 4 ( - 3650.45175 \dots ) \cdot 3443.06627 \dots}{2 ( - 3650.45175 \dots )}

(- 191.076)^{2} - 4 (- 3650.45175 \dots) \cdot 3443.06627 \dots = 50311499.28372 \dots

(- 191.076)^{2} - 4 (- 3650.45175 \dots) \cdot 3443.06627 \dots

Apply rule

- (- a) = a

= (- 191.076)^{2} + 4 \cdot 3650.45175 \dots \cdot 3443.06627 \dots

Apply exponent rule:

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

n

is even

(- 191.076)^{2} = 191.07 6^{2}

= 191.07 6^{2} + 4 \cdot 3443.06627 \dots \cdot 3650.45175 \dots

Multiply the numbers:

4 \cdot 3650.45175 \dots \cdot 3443.06627 \dots = 50274989.24594 \dots

= 191.07 6^{2} + 50274989.24594 \dots

191.07 6^{2} = 36510.037776

= 36510.037776 + 50274989.24594 \dots

Add the numbers:

36510.037776 + 50274989.24594 \dots = 50311499.28372 \dots

= 50311499.28372 \dots

u_{1, 2} = \frac{- ( - 191.076 ) \pm 50311499.28372 \dots}{2 ( - 3650.45175 \dots )}

Separate the solutions

u_{1} = \frac{- ( - 191.076 ) + 50311499.28372 \dots}{2 ( - 3650.45175 \dots )}, u_{2} = \frac{- ( - 191.076 ) - 50311499.28372 \dots}{2 ( - 3650.45175 \dots )}

u = \frac{- ( - 191.076 ) + 50311499.28372 \dots}{2 ( - 3650.45175 \dots )} : - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}

\frac{- ( - 191.076 ) + 50311499.28372 \dots}{2 ( - 3650.45175 \dots )}

Remove parentheses:

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

= \frac{191.076 + 50311499.28372 \dots}{- 2 \cdot 3650.45175 \dots}

Multiply the numbers:

2 \cdot 3650.45175 \dots = 7300.90350 \dots

= \frac{191.076 + 50311499.28372 \dots}{- 7300.90350 \dots}

Apply the fraction rule:

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

= - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}

u = \frac{- ( - 191.076 ) - 50311499.28372 \dots}{2 ( - 3650.45175 \dots )} : \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

\frac{- ( - 191.076 ) - 50311499.28372 \dots}{2 ( - 3650.45175 \dots )}

Remove parentheses:

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

= \frac{191.076 - 50311499.28372 \dots}{- 2 \cdot 3650.45175 \dots}

Multiply the numbers:

2 \cdot 3650.45175 \dots = 7300.90350 \dots

= \frac{191.076 - 50311499.28372 \dots}{- 7300.90350 \dots}

Apply the fraction rule:

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

191.076 - 50311499.28372 \dots = - (50311499.28372 \dots - 191.076)

= \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

The solutions to the quadratic equation are:

u = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}, u = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

Substitute back

u = sin (x)

sin (x) = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}, sin (x) = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

sin (x) = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}, sin (x) = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

sin (x) = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots} : x = arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn

sin (x) = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}

Apply trig inverse properties

sin (x) = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}

General solutions for

sin (x) = - \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}

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

x = arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn

x = arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn

sin (x) = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots} : x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn, x = π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

sin (x) = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

Apply trig inverse properties

sin (x) = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

General solutions for

sin (x) = \frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}

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

x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn, x = π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn, x = π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn, x = π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

59.2153 cos (x) - 12 sin (x) = 7.9615

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

Check the solution

arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn :

False

arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn

Plug in

n = 1

arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 π 1

For

59.2153 cos (x) - 12 sin (x) = 7.9615

plug in

x = arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 π 1

59.2153 cos (arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 π 1) - 12 sin (arcsin (- \frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 π 1) = 7.9615

Refine

15.98338 \dots = 7.9615

\Rightarrow False

Check the solution

π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn :

True

π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 π 1

For

59.2153 cos (x) - 12 sin (x) = 7.9615

plug in

x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 π 1

59.2153 cos (π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 π 1) - 12 sin (π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 π 1) = 7.9615

Refine

7.96149 \dots = 7.9615

\Rightarrow True

Check the solution

arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn :

True

arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

Plug in

n = 1

arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 π 1

For

59.2153 cos (x) - 12 sin (x) = 7.9615

plug in

x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 π 1

59.2153 cos (arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 π 1) - 12 sin (arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 π 1) = 7.9615

Refine

7.9615 = 7.9615

\Rightarrow True

Check the solution

π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn :

False

π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 π 1

For

59.2153 cos (x) - 12 sin (x) = 7.9615

plug in

x = π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 π 1

59.2153 cos (π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 π 1) - 12 sin (π - arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 π 1) = 7.9615

Refine

- 30.65014 \dots = 7.9615

\Rightarrow False

x = π + arcsin (\frac{191.076 + 50311499.28372 \dots}{7300.90350 \dots}) + 2 πn, x = arcsin (\frac{50311499.28372 \dots - 191.076}{7300.90350 \dots}) + 2 πn

Show solutions in decimal form

x = π + 1.50300 \dots + 2 πn, x = 1.23869 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 7.9615=59.2153cos(x)-12sin(x) ?
The general solution for 7.9615=59.2153cos(x)-12sin(x) is x=pi+1.50300…+2pin,x=1.23869…+2pin