What is the general solution for 117.72sin(θ)-35.316cos(θ)-12.5=0 ?

The general solution for 117.72sin(θ)-35.316cos(θ)-12.5=0 is θ=0.39333…+2pin,θ=pi+0.18957…+2pin

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117.72sin(θ)-35.316cos(θ)-12.5=0

Solution

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

Solution

θ = 0.39333 \dots + 2 πn, θ = π + 0.18957 \dots + 2 πn

Degrees

θ = 22.53666 \dots^{\circ} + 36 0^{\circ} n, θ = 190.86182 \dots^{\circ} + 36 0^{\circ} n

Solution steps

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

Add

35.316 cos (θ)

to both sides

117.72 sin (θ) - 12.5 = 35.316 cos (θ)

Square both sides

(117.72 sin (θ) - 12.5)^{2} = (35.316 cos (θ))^{2}

Subtract

(35.316 cos (θ))^{2}

from both sides

(117.72 sin (θ) - 12.5)^{2} - 1247.219856 cos^{2} (θ) = 0

Rewrite using trig identities

(- 12.5 + 117.72 sin (θ))^{2} - 1247.219856 cos^{2} (θ)

Use the Pythagorean identity:

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

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

= (- 12.5 + 117.72 sin (θ))^{2} - 1247.219856 (1 - sin^{2} (θ))

Simplify

(- 12.5 + 117.72 sin (θ))^{2} - 1247.219856 (1 - sin^{2} (θ)) : 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

(- 12.5 + 117.72 sin (θ))^{2} - 1247.219856 (1 - sin^{2} (θ))

(- 12.5 + 117.72 sin (θ))^{2} : 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ)

Apply Perfect Square Formula:

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

a = - 12.5, b = 117.72 sin (θ)

= (- 12.5)^{2} + 2 (- 12.5) \cdot 117.72 sin (θ) + (117.72 sin (θ))^{2}

Simplify

(- 12.5)^{2} + 2 (- 12.5) \cdot 117.72 sin (θ) + (117.72 sin (θ))^{2} : 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ)

(- 12.5)^{2} + 2 (- 12.5) \cdot 117.72 sin (θ) + (117.72 sin (θ))^{2}

Remove parentheses:

(- a) = - a

= (- 12.5)^{2} - 2 \cdot 12.5 \cdot 117.72 sin (θ) + (117.72 sin (θ))^{2}

(- 12.5)^{2} = 156.25

(- 12.5)^{2}

Apply exponent rule:

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

n

is even

(- 12.5)^{2} = 12. 5^{2}

= 12. 5^{2}

12. 5^{2} = 156.25

= 156.25

2 \cdot 12.5 \cdot 117.72 sin (θ) = 2943 sin (θ)

2 \cdot 12.5 \cdot 117.72 sin (θ)

Multiply the numbers:

2 \cdot 12.5 \cdot 117.72 = 2943

= 2943 sin (θ)

(117.72 sin (θ))^{2} = 13857.9984 sin^{2} (θ)

(117.72 sin (θ))^{2}

Apply exponent rule:

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

= 117.7 2^{2} sin^{2} (θ)

117.7 2^{2} = 13857.9984

= 13857.9984 sin^{2} (θ)

= 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ)

= 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ)

= 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ) - 1247.219856 (1 - sin^{2} (θ))

Expand

- 1247.219856 (1 - sin^{2} (θ)) : - 1247.219856 + 1247.219856 sin^{2} (θ)

- 1247.219856 (1 - sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = - 1247.219856, b = 1, c = sin^{2} (θ)

= - 1247.219856 \cdot 1 - (- 1247.219856) sin^{2} (θ)

Apply minus-plus rules

- (- a) = a

= - 1 \cdot 1247.219856 + 1247.219856 sin^{2} (θ)

Multiply the numbers:

1 \cdot 1247.219856 = 1247.219856

= - 1247.219856 + 1247.219856 sin^{2} (θ)

= 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ) - 1247.219856 + 1247.219856 sin^{2} (θ)

Simplify

156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ) - 1247.219856 + 1247.219856 sin^{2} (θ) : 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ) - 1247.219856 + 1247.219856 sin^{2} (θ)

Group like terms

= - 2943 sin (θ) + 13857.9984 sin^{2} (θ) + 1247.219856 sin^{2} (θ) + 156.25 - 1247.219856

Add similar elements:

13857.9984 sin^{2} (θ) + 1247.219856 sin^{2} (θ) = 15105.218256 sin^{2} (θ)

= - 2943 sin (θ) + 15105.218256 sin^{2} (θ) + 156.25 - 1247.219856

Add/Subtract the numbers:

156.25 - 1247.219856 = - 1090.969856

= 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

= 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

= 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

- 1090.969856 + 15105.218256 sin^{2} (θ) - 2943 sin (θ) = 0

Solve by substitution

- 1090.969856 + 15105.218256 sin^{2} (θ) - 2943 sin (θ) = 0

Let:

sin (θ) = u

- 1090.969856 + 15105.218256 u^{2} - 2943 u = 0

- 1090.969856 + 15105.218256 u^{2} - 2943 u = 0 : u = \frac{0.19483 \dots + 0.32685 \dots}{2}, u = \frac{0.19483 \dots - 0.32685 \dots}{2}

- 1090.969856 + 15105.218256 u^{2} - 2943 u = 0

Divide both sides by

15105.218256

- \frac{1090.969856}{15105.218256} + \frac{15105.218256 u ^{2}}{15105.218256} - \frac{2943 u}{15105.218256} = \frac{0}{15105.218256}

Write in the standard form

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

u^{2} - 0.19483 \dots u - 0.07222 \dots = 0

Solve with the quadratic formula

u^{2} - 0.19483 \dots u - 0.07222 \dots = 0

Quadratic Equation Formula:

For

a = 1, b = - 0.19483 \dots, c = - 0.07222 \dots

u_{1, 2} = \frac{- ( - 0.19483 \dots ) \pm ( - 0.19483 \dots ) ^{2} - 4 \cdot 1 \cdot ( - 0.07222 \dots )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 0.19483 \dots ) \pm ( - 0.19483 \dots ) ^{2} - 4 \cdot 1 \cdot ( - 0.07222 \dots )}{2 \cdot 1}

(- 0.19483 \dots)^{2} - 4 \cdot 1 \cdot (- 0.07222 \dots) = 0.32685 \dots

(- 0.19483 \dots)^{2} - 4 \cdot 1 \cdot (- 0.07222 \dots)

Apply rule

- (- a) = a

= (- 0.19483 \dots)^{2} + 4 \cdot 1 \cdot 0.07222 \dots

Apply exponent rule:

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

n

is even

(- 0.19483 \dots)^{2} = 0.19483 \dots^{2}

= 0.19483 \dots^{2} + 4 \cdot 1 \cdot 0.07222 \dots

Multiply the numbers:

4 \cdot 1 \cdot 0.07222 \dots = 0.28889 \dots

= 0.19483 \dots^{2} + 0.28889 \dots

0.19483 \dots^{2} = 0.03796 \dots

= 0.03796 \dots + 0.28889 \dots

Add the numbers:

0.03796 \dots + 0.28889 \dots = 0.32685 \dots

= 0.32685 \dots

u_{1, 2} = \frac{- ( - 0.19483 \dots ) \pm 0.32685 \dots}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 0.19483 \dots ) + 0.32685 \dots}{2 \cdot 1}, u_{2} = \frac{- ( - 0.19483 \dots ) - 0.32685 \dots}{2 \cdot 1}

u = \frac{- ( - 0.19483 \dots ) + 0.32685 \dots}{2 \cdot 1} : \frac{0.19483 \dots + 0.32685 \dots}{2}

\frac{- ( - 0.19483 \dots ) + 0.32685 \dots}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{0.19483 \dots + 0.32685 \dots}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0.19483 \dots + 0.32685 \dots}{2}

u = \frac{- ( - 0.19483 \dots ) - 0.32685 \dots}{2 \cdot 1} : \frac{0.19483 \dots - 0.32685 \dots}{2}

\frac{- ( - 0.19483 \dots ) - 0.32685 \dots}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{0.19483 \dots - 0.32685 \dots}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0.19483 \dots - 0.32685 \dots}{2}

The solutions to the quadratic equation are:

u = \frac{0.19483 \dots + 0.32685 \dots}{2}, u = \frac{0.19483 \dots - 0.32685 \dots}{2}

Substitute back

u = sin (θ)

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}, sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}, sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2} : θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}

Apply trig inverse properties

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}

General solutions for

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}

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

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2} : θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

Apply trig inverse properties

sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

General solutions for

sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

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

θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

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

Check the solution

arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn :

True

arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

Plug in

n = 1

arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1

For

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

plug in

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1

117.72 sin (arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1) - 35.316 cos (arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1) - 12.5 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn :

False

π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1

For

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

plug in

θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1

117.72 sin (π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1) - 35.316 cos (π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1) - 12.5 = 0

Refine

65.23815 \dots = 0

\Rightarrow False

Check the solution

arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn :

False

arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Plug in

n = 1

arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1

For

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

plug in

θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1

117.72 sin (arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1) - 35.316 cos (arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1) - 12.5 = 0

Refine

- 69.36659 \dots = 0

\Rightarrow False

Check the solution

π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn :

True

π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Plug in

n = 1

π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1

For

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

plug in

θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1

117.72 sin (π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1) - 35.316 cos (π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1) - 12.5 = 0

Refine

0 = 0

\Rightarrow True

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Show solutions in decimal form

θ = 0.39333 \dots + 2 πn, θ = π + 0.18957 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 117.72sin(θ)-35.316cos(θ)-12.5=0 ?
The general solution for 117.72sin(θ)-35.316cos(θ)-12.5=0 is θ=0.39333…+2pin,θ=pi+0.18957…+2pin