What is the general solution for 88.2sin(x)-12.78=0.1*88.2cos(x) ?

The general solution for 88.2sin(x)-12.78=0.1*88.2cos(x) is x=0.24435…+2pin,x=pi-0.04501…+2pin

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88.2sin(x)-12.78=0.1*88.2cos(x)

Solution

88.2 sin (x) - 12.78 = 0.1 \cdot 88.2 cos (x)

Solution

x = 0.24435 \dots + 2 πn, x = π - 0.04501 \dots + 2 πn

Degrees

x = 14.00032 \dots^{\circ} + 36 0^{\circ} n, x = 177.42086 \dots^{\circ} + 36 0^{\circ} n

Solution steps

88.2 sin (x) - 12.78 = 0.1 \cdot 88.2 cos (x)

Square both sides

(88.2 sin (x) - 12.78)^{2} = (0.1 \cdot 88.2 cos (x))^{2}

Subtract

(0.188.2 cos (x))^{2}

from both sides

(88.2 sin (x) - 12.78)^{2} - 77.7924 cos^{2} (x) = 0

Rewrite using trig identities

(- 12.78 + 88.2 sin (x))^{2} - 77.7924 cos^{2} (x)

Use the Pythagorean identity:

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

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

= (- 12.78 + 88.2 sin (x))^{2} - 77.7924 (1 - sin^{2} (x))

Simplify

(- 12.78 + 88.2 sin (x))^{2} - 77.7924 (1 - sin^{2} (x)) : 7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

(- 12.78 + 88.2 sin (x))^{2} - 77.7924 (1 - sin^{2} (x))

(- 12.78 + 88.2 sin (x))^{2} : 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = - 12.78, b = 88.2 sin (x)

= (- 12.78)^{2} + 2 (- 12.78) \cdot 88.2 sin (x) + (88.2 sin (x))^{2}

Simplify

(- 12.78)^{2} + 2 (- 12.78) \cdot 88.2 sin (x) + (88.2 sin (x))^{2} : 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x)

(- 12.78)^{2} + 2 (- 12.78) \cdot 88.2 sin (x) + (88.2 sin (x))^{2}

Remove parentheses:

(- a) = - a

= (- 12.78)^{2} - 2 \cdot 12.78 \cdot 88.2 sin (x) + (88.2 sin (x))^{2}

(- 12.78)^{2} = 163.3284

(- 12.78)^{2}

Apply exponent rule:

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

n

is even

(- 12.78)^{2} = 12.7 8^{2}

= 12.7 8^{2}

12.7 8^{2} = 163.3284

= 163.3284

2 \cdot 12.78 \cdot 88.2 sin (x) = 2254.392 sin (x)

2 \cdot 12.78 \cdot 88.2 sin (x)

Multiply the numbers:

2 \cdot 12.78 \cdot 88.2 = 2254.392

= 2254.392 sin (x)

(88.2 sin (x))^{2} = 7779.24 sin^{2} (x)

(88.2 sin (x))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 88. 2^{2} sin^{2} (x)

88. 2^{2} = 7779.24

= 7779.24 sin^{2} (x)

= 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x)

= 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x)

= 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x) - 77.7924 (1 - sin^{2} (x))

Expand

- 77.7924 (1 - sin^{2} (x)) : - 77.7924 + 77.7924 sin^{2} (x)

- 77.7924 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 77.7924, b = 1, c = sin^{2} (x)

= - 77.7924 \cdot 1 - (- 77.7924) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 1 \cdot 77.7924 + 77.7924 sin^{2} (x)

Multiply the numbers:

1 \cdot 77.7924 = 77.7924

= - 77.7924 + 77.7924 sin^{2} (x)

= 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x) - 77.7924 + 77.7924 sin^{2} (x)

Simplify

163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x) - 77.7924 + 77.7924 sin^{2} (x) : 7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x) - 77.7924 + 77.7924 sin^{2} (x)

Group like terms

= - 2254.392 sin (x) + 7779.24 sin^{2} (x) + 77.7924 sin^{2} (x) + 163.3284 - 77.7924

Add similar elements:

7779.24 sin^{2} (x) + 77.7924 sin^{2} (x) = 7857.0324 sin^{2} (x)

= - 2254.392 sin (x) + 7857.0324 sin^{2} (x) + 163.3284 - 77.7924

Add/Subtract the numbers:

163.3284 - 77.7924 = 85.536

= 7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

= 7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

= 7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

85.536 - 2254.392 sin (x) + 7857.0324 sin^{2} (x) = 0

Solve by substitution

85.536 - 2254.392 sin (x) + 7857.0324 sin^{2} (x) = 0

Let:

sin (x) = u

85.536 - 2254.392 u + 7857.0324 u^{2} = 0

85.536 - 2254.392 u + 7857.0324 u^{2} = 0 : u = \frac{0.28692 \dots + 0.03878 \dots}{2}, u = \frac{0.28692 \dots - 0.03878 \dots}{2}

85.536 - 2254.392 u + 7857.0324 u^{2} = 0

Divide both sides by

7857.0324

\frac{85.536}{7857.0324} - \frac{2254.392 u}{7857.0324} + \frac{7857.0324 u ^{2}}{7857.0324} = \frac{0}{7857.0324}

Write in the standard form

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

u^{2} - 0.28692 \dots u + 0.01088 \dots = 0

Solve with the quadratic formula

u^{2} - 0.28692 \dots u + 0.01088 \dots = 0

Quadratic Equation Formula:

For

a = 1, b = - 0.28692 \dots, c = 0.01088 \dots

u_{1, 2} = \frac{- ( - 0.28692 \dots ) \pm ( - 0.28692 \dots ) ^{2} - 4 \cdot 1 \cdot 0.01088 \dots}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 0.28692 \dots ) \pm ( - 0.28692 \dots ) ^{2} - 4 \cdot 1 \cdot 0.01088 \dots}{2 \cdot 1}

(- 0.28692 \dots)^{2} - 4 \cdot 1 \cdot 0.01088 \dots = 0.03878 \dots

(- 0.28692 \dots)^{2} - 4 \cdot 1 \cdot 0.01088 \dots

Apply exponent rule:

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

n

is even

(- 0.28692 \dots)^{2} = 0.28692 \dots^{2}

= 0.28692 \dots^{2} - 4 \cdot 1 \cdot 0.01088 \dots

Multiply the numbers:

4 \cdot 1 \cdot 0.01088 \dots = 0.04354 \dots

= 0.28692 \dots^{2} - 0.04354 \dots

0.28692 \dots^{2} = 0.08232 \dots

= 0.08232 \dots - 0.04354 \dots

Subtract the numbers:

0.08232 \dots - 0.04354 \dots = 0.03878 \dots

= 0.03878 \dots

u_{1, 2} = \frac{- ( - 0.28692 \dots ) \pm 0.03878 \dots}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 0.28692 \dots ) + 0.03878 \dots}{2 \cdot 1}, u_{2} = \frac{- ( - 0.28692 \dots ) - 0.03878 \dots}{2 \cdot 1}

u = \frac{- ( - 0.28692 \dots ) + 0.03878 \dots}{2 \cdot 1} : \frac{0.28692 \dots + 0.03878 \dots}{2}

\frac{- ( - 0.28692 \dots ) + 0.03878 \dots}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{0.28692 \dots + 0.03878 \dots}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0.28692 \dots + 0.03878 \dots}{2}

u = \frac{- ( - 0.28692 \dots ) - 0.03878 \dots}{2 \cdot 1} : \frac{0.28692 \dots - 0.03878 \dots}{2}

\frac{- ( - 0.28692 \dots ) - 0.03878 \dots}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{0.28692 \dots - 0.03878 \dots}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0.28692 \dots - 0.03878 \dots}{2}

The solutions to the quadratic equation are:

u = \frac{0.28692 \dots + 0.03878 \dots}{2}, u = \frac{0.28692 \dots - 0.03878 \dots}{2}

Substitute back

u = sin (x)

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}, sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}, sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2} : x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}

Apply trig inverse properties

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}

General solutions for

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}

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

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2} : x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

Apply trig inverse properties

sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

General solutions for

sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

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

x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

Combine all the solutions

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

88.2 sin (x) - 12.78 = 0.188.2 cos (x)

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

Check the solution

arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn :

True

arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

Plug in

n = 1

arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1

For

88.2 sin (x) - 12.78 = 0.188.2 cos (x)

plug in

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1

88.2 sin (arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1) - 12.78 = 0.1 \cdot 88.2 cos (arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1)

Refine

8.55799 \dots = 8.55799 \dots

\Rightarrow True

Check the solution

π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn :

False

π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1

For

88.2 sin (x) - 12.78 = 0.188.2 cos (x)

plug in

x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1

88.2 sin (π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1) - 12.78 = 0.1 \cdot 88.2 cos (π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1)

Refine

8.55799 \dots = - 8.55799 \dots

\Rightarrow False

Check the solution

arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn :

False

arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

Plug in

n = 1

arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1

For

88.2 sin (x) - 12.78 = 0.188.2 cos (x)

plug in

x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1

88.2 sin (arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1) - 12.78 = 0.1 \cdot 88.2 cos (arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1)

Refine

- 8.81106 \dots = 8.81106 \dots

\Rightarrow False

Check the solution

π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn :

True

π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1

For

88.2 sin (x) - 12.78 = 0.188.2 cos (x)

plug in

x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1

88.2 sin (π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1) - 12.78 = 0.1 \cdot 88.2 cos (π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1)

Refine

- 8.81106 \dots = - 8.81106 \dots

\Rightarrow True

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

Show solutions in decimal form

x = 0.24435 \dots + 2 πn, x = π - 0.04501 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 88.2sin(x)-12.78=0.1*88.2cos(x) ?
The general solution for 88.2sin(x)-12.78=0.1*88.2cos(x) is x=0.24435…+2pin,x=pi-0.04501…+2pin