What is the general solution for sin(θ)-0.2cos(θ)=(8.54)/(9.8) ?

The general solution for sin(θ)-0.2cos(θ)=(8.54)/(9.8) is θ=2.31438…+2pin,θ=1.22199…+2pin

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sin(θ)-0.2cos(θ)=(8.54)/(9.8)

Solution

sin (θ) - 0.2 cos (θ) = \frac{8.54}{9.8}

Solution

θ = 2.31438 \dots + 2 πn, θ = 1.22199 \dots + 2 πn

Degrees

θ = 132.60472 \dots^{\circ} + 36 0^{\circ} n, θ = 70.01513 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ) - 0.2 cos (θ) = \frac{8.54}{9.8}

Add

0.2 cos (θ)

to both sides

sin (θ) = 0.87142 \dots + 0.2 cos (θ)

Square both sides

sin^{2} (θ) = (0.87142 \dots + 0.2 cos (θ))^{2}

Subtract

(0.87142 \dots + 0.2 cos (θ))^{2}

from both sides

sin^{2} (θ) - 0.75938 \dots - 0.34857 \dots cos (θ) - 0.04 cos^{2} (θ) = 0

Rewrite using trig identities

- 0.75938 \dots + sin^{2} (θ) - 0.04 cos^{2} (θ) - 0.34857 \dots cos (θ)

Use the Pythagorean identity:

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

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

= - 0.75938 \dots + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.34857 \dots cos (θ)

Simplify

- 0.75938 \dots + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.34857 \dots cos (θ) : - 1.04 cos^{2} (θ) - 0.34857 \dots cos (θ) + 0.24061 \dots

- 0.75938 \dots + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.34857 \dots cos (θ)

Add similar elements:

- cos^{2} (θ) - 0.04 cos^{2} (θ) = - 1.04 cos^{2} (θ)

= - 0.75938 \dots + 1 - 1.04 cos^{2} (θ) - 0.34857 \dots cos (θ)

Add/Subtract the numbers:

- 0.75938 \dots + 1 = 0.24061 \dots

= - 1.04 cos^{2} (θ) - 0.34857 \dots cos (θ) + 0.24061 \dots

= - 1.04 cos^{2} (θ) - 0.34857 \dots cos (θ) + 0.24061 \dots

0.24061 \dots - 0.34857 \dots cos (θ) - 1.04 cos^{2} (θ) = 0

Solve by substitution

0.24061 \dots - 0.34857 \dots cos (θ) - 1.04 cos^{2} (θ) = 0

Let:

cos (θ) = u

0.24061 \dots - 0.34857 \dots u - 1.04 u^{2} = 0

0.24061 \dots - 0.34857 \dots u - 1.04 u^{2} = 0 : u = - \frac{0.34857 \dots + 1.12244 \dots}{2.08}, u = \frac{1.12244 \dots - 0.34857 \dots}{2.08}

0.24061 \dots - 0.34857 \dots u - 1.04 u^{2} = 0

Write in the standard form

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

- 1.04 u^{2} - 0.34857 \dots u + 0.24061 \dots = 0

Solve with the quadratic formula

- 1.04 u^{2} - 0.34857 \dots u + 0.24061 \dots = 0

Quadratic Equation Formula:

For

a = - 1.04, b = - 0.34857 \dots, c = 0.24061 \dots

u_{1, 2} = \frac{- ( - 0.34857 \dots ) \pm ( - 0.34857 \dots ) ^{2} - 4 ( - 1.04 ) \cdot 0.24061 \dots}{2 ( - 1.04 )}

u_{1, 2} = \frac{- ( - 0.34857 \dots ) \pm ( - 0.34857 \dots ) ^{2} - 4 ( - 1.04 ) \cdot 0.24061 \dots}{2 ( - 1.04 )}

(- 0.34857 \dots)^{2} - 4 (- 1.04) \cdot 0.24061 \dots = 1.12244 \dots

(- 0.34857 \dots)^{2} - 4 (- 1.04) \cdot 0.24061 \dots

Apply rule

- (- a) = a

= (- 0.34857 \dots)^{2} + 4 \cdot 1.04 \cdot 0.24061 \dots

Apply exponent rule:

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

n

is even

(- 0.34857 \dots)^{2} = 0.34857 \dots^{2}

= 0.34857 \dots^{2} + 4 \cdot 0.24061 \dots \cdot 1.04

Multiply the numbers:

4 \cdot 1.04 \cdot 0.24061 \dots = 1.00094 \dots

= 0.34857 \dots^{2} + 1.00094 \dots

0.34857 \dots^{2} = 0.12150 \dots

= 0.12150 \dots + 1.00094 \dots

Add the numbers:

0.12150 \dots + 1.00094 \dots = 1.12244 \dots

= 1.12244 \dots

u_{1, 2} = \frac{- ( - 0.34857 \dots ) \pm 1.12244 \dots}{2 ( - 1.04 )}

Separate the solutions

u_{1} = \frac{- ( - 0.34857 \dots ) + 1.12244 \dots}{2 ( - 1.04 )}, u_{2} = \frac{- ( - 0.34857 \dots ) - 1.12244 \dots}{2 ( - 1.04 )}

u = \frac{- ( - 0.34857 \dots ) + 1.12244 \dots}{2 ( - 1.04 )} : - \frac{0.34857 \dots + 1.12244 \dots}{2.08}

\frac{- ( - 0.34857 \dots ) + 1.12244 \dots}{2 ( - 1.04 )}

Remove parentheses:

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

= \frac{0.34857 \dots + 1.12244 \dots}{- 2 \cdot 1.04}

Multiply the numbers:

2 \cdot 1.04 = 2.08

= \frac{0.34857 \dots + 1.12244 \dots}{- 2.08}

Apply the fraction rule:

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

= - \frac{0.34857 \dots + 1.12244 \dots}{2.08}

u = \frac{- ( - 0.34857 \dots ) - 1.12244 \dots}{2 ( - 1.04 )} : \frac{1.12244 \dots - 0.34857 \dots}{2.08}

\frac{- ( - 0.34857 \dots ) - 1.12244 \dots}{2 ( - 1.04 )}

Remove parentheses:

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

= \frac{0.34857 \dots - 1.12244 \dots}{- 2 \cdot 1.04}

Multiply the numbers:

2 \cdot 1.04 = 2.08

= \frac{0.34857 \dots - 1.12244 \dots}{- 2.08}

Apply the fraction rule:

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

0.34857 \dots - 1.12244 \dots = - (1.12244 \dots - 0.34857 \dots)

= \frac{1.12244 \dots - 0.34857 \dots}{2.08}

The solutions to the quadratic equation are:

u = - \frac{0.34857 \dots + 1.12244 \dots}{2.08}, u = \frac{1.12244 \dots - 0.34857 \dots}{2.08}

Substitute back

u = cos (θ)

cos (θ) = - \frac{0.34857 \dots + 1.12244 \dots}{2.08}, cos (θ) = \frac{1.12244 \dots - 0.34857 \dots}{2.08}

cos (θ) = - \frac{0.34857 \dots + 1.12244 \dots}{2.08}, cos (θ) = \frac{1.12244 \dots - 0.34857 \dots}{2.08}

cos (θ) = - \frac{0.34857 \dots + 1.12244 \dots}{2.08} : θ = arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn, θ = - arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn

cos (θ) = - \frac{0.34857 \dots + 1.12244 \dots}{2.08}

Apply trig inverse properties

cos (θ) = - \frac{0.34857 \dots + 1.12244 \dots}{2.08}

General solutions for

cos (θ) = - \frac{0.34857 \dots + 1.12244 \dots}{2.08}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

θ = arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn, θ = - arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn

θ = arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn, θ = - arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn

cos (θ) = \frac{1.12244 \dots - 0.34857 \dots}{2.08} : θ = arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn

cos (θ) = \frac{1.12244 \dots - 0.34857 \dots}{2.08}

Apply trig inverse properties

cos (θ) = \frac{1.12244 \dots - 0.34857 \dots}{2.08}

General solutions for

cos (θ) = \frac{1.12244 \dots - 0.34857 \dots}{2.08}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

θ = arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn

θ = arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn

Combine all the solutions

θ = arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn, θ = - arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn, θ = arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sin (θ) - 0.2 cos (θ) = \frac{8.54}{9.8}

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

Check the solution

arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn :

True

arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn

Plug in

n = 1

arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 π 1

For

sin (θ) - 0.2 cos (θ) = \frac{8.54}{9.8}

plug in

θ = arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 π 1

sin (arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 π 1) - 0.2 cos (arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 π 1) = \frac{8.54}{9.8}

Refine

0.87142 \dots = 0.87142 \dots

\Rightarrow True

Check the solution

- arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn :

False

- arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn

Plug in

n = 1

- arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 π 1

For

sin (θ) - 0.2 cos (θ) = \frac{8.54}{9.8}

plug in

θ = - arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 π 1

sin (- arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 π 1) - 0.2 cos (- arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 π 1) = \frac{8.54}{9.8}

Refine

- 0.60065 \dots = 0.87142 \dots

\Rightarrow False

Check the solution

arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn :

True

arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn

Plug in

n = 1

arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 π 1

For

sin (θ) - 0.2 cos (θ) = \frac{8.54}{9.8}

plug in

θ = arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 π 1

sin (arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 π 1) - 0.2 cos (arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 π 1) = \frac{8.54}{9.8}

Refine

0.87142 \dots = 0.87142 \dots

\Rightarrow True

Check the solution

2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn :

False

2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 π 1

For

sin (θ) - 0.2 cos (θ) = \frac{8.54}{9.8}

plug in

θ = 2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 π 1

sin (2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 π 1) - 0.2 cos (2 π - arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 π 1) = \frac{8.54}{9.8}

Refine

- 1.00813 \dots = 0.87142 \dots

\Rightarrow False

θ = arccos (- \frac{0.34857 \dots + 1.12244 \dots}{2.08}) + 2 πn, θ = arccos (\frac{1.12244 \dots - 0.34857 \dots}{2.08}) + 2 πn

Show solutions in decimal form

θ = 2.31438 \dots + 2 πn, θ = 1.22199 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)-0.2cos(θ)=(8.54)/(9.8) ?
The general solution for sin(θ)-0.2cos(θ)=(8.54)/(9.8) is θ=2.31438…+2pin,θ=1.22199…+2pin