What is the general solution for sin(θ)-0.1cos(θ)=(8.87)/(9.8) ?

The general solution for sin(θ)-0.1cos(θ)=(8.87)/(9.8) is θ=2.12008…+2pin,θ=1.22084…+2pin

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sin(θ)-0.1cos(θ)=(8.87)/(9.8)

Solution

sin (θ) - 0.1 cos (θ) = \frac{8.87}{9.8}

Solution

θ = 2.12008 \dots + 2 πn, θ = 1.22084 \dots + 2 πn

Degrees

θ = 121.47220 \dots^{\circ} + 36 0^{\circ} n, θ = 69.94898 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ) - 0.1 cos (θ) = \frac{8.87}{9.8}

Add

0.1 cos (θ)

to both sides

sin (θ) = 0.90510 \dots + 0.1 cos (θ)

Square both sides

sin^{2} (θ) = (0.90510 \dots + 0.1 cos (θ))^{2}

Subtract

(0.90510 \dots + 0.1 cos (θ))^{2}

from both sides

sin^{2} (θ) - 0.81920 \dots - 0.18102 \dots cos (θ) - 0.01 cos^{2} (θ) = 0

Rewrite using trig identities

- 0.81920 \dots + sin^{2} (θ) - 0.01 cos^{2} (θ) - 0.18102 \dots cos (θ)

Use the Pythagorean identity:

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

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

= - 0.81920 \dots + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.18102 \dots cos (θ)

Simplify

- 0.81920 \dots + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.18102 \dots cos (θ) : - 1.01 cos^{2} (θ) - 0.18102 \dots cos (θ) + 0.18079 \dots

- 0.81920 \dots + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.18102 \dots cos (θ)

Add similar elements:

- cos^{2} (θ) - 0.01 cos^{2} (θ) = - 1.01 cos^{2} (θ)

= - 0.81920 \dots + 1 - 1.01 cos^{2} (θ) - 0.18102 \dots cos (θ)

Add/Subtract the numbers:

- 0.81920 \dots + 1 = 0.18079 \dots

= - 1.01 cos^{2} (θ) - 0.18102 \dots cos (θ) + 0.18079 \dots

= - 1.01 cos^{2} (θ) - 0.18102 \dots cos (θ) + 0.18079 \dots

0.18079 \dots - 0.18102 \dots cos (θ) - 1.01 cos^{2} (θ) = 0

Solve by substitution

0.18079 \dots - 0.18102 \dots cos (θ) - 1.01 cos^{2} (θ) = 0

Let:

cos (θ) = u

0.18079 \dots - 0.18102 \dots u - 1.01 u^{2} = 0

0.18079 \dots - 0.18102 \dots u - 1.01 u^{2} = 0 : u = - \frac{0.18102 \dots + 0.76316 \dots}{2.02}, u = \frac{0.76316 \dots - 0.18102 \dots}{2.02}

0.18079 \dots - 0.18102 \dots u - 1.01 u^{2} = 0

Write in the standard form

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

- 1.01 u^{2} - 0.18102 \dots u + 0.18079 \dots = 0

Solve with the quadratic formula

- 1.01 u^{2} - 0.18102 \dots u + 0.18079 \dots = 0

Quadratic Equation Formula:

For

a = - 1.01, b = - 0.18102 \dots, c = 0.18079 \dots

u_{1, 2} = \frac{- ( - 0.18102 \dots ) \pm ( - 0.18102 \dots ) ^{2} - 4 ( - 1.01 ) \cdot 0.18079 \dots}{2 ( - 1.01 )}

u_{1, 2} = \frac{- ( - 0.18102 \dots ) \pm ( - 0.18102 \dots ) ^{2} - 4 ( - 1.01 ) \cdot 0.18079 \dots}{2 ( - 1.01 )}

(- 0.18102 \dots)^{2} - 4 (- 1.01) \cdot 0.18079 \dots = 0.76316 \dots

(- 0.18102 \dots)^{2} - 4 (- 1.01) \cdot 0.18079 \dots

Apply rule

- (- a) = a

= (- 0.18102 \dots)^{2} + 4 \cdot 1.01 \cdot 0.18079 \dots

Apply exponent rule:

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

n

is even

(- 0.18102 \dots)^{2} = 0.18102 \dots^{2}

= 0.18102 \dots^{2} + 4 \cdot 0.18079 \dots \cdot 1.01

Multiply the numbers:

4 \cdot 1.01 \cdot 0.18079 \dots = 0.73039 \dots

= 0.18102 \dots^{2} + 0.73039 \dots

0.18102 \dots^{2} = 0.03276 \dots

= 0.03276 \dots + 0.73039 \dots

Add the numbers:

0.03276 \dots + 0.73039 \dots = 0.76316 \dots

= 0.76316 \dots

u_{1, 2} = \frac{- ( - 0.18102 \dots ) \pm 0.76316 \dots}{2 ( - 1.01 )}

Separate the solutions

u_{1} = \frac{- ( - 0.18102 \dots ) + 0.76316 \dots}{2 ( - 1.01 )}, u_{2} = \frac{- ( - 0.18102 \dots ) - 0.76316 \dots}{2 ( - 1.01 )}

u = \frac{- ( - 0.18102 \dots ) + 0.76316 \dots}{2 ( - 1.01 )} : - \frac{0.18102 \dots + 0.76316 \dots}{2.02}

\frac{- ( - 0.18102 \dots ) + 0.76316 \dots}{2 ( - 1.01 )}

Remove parentheses:

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

= \frac{0.18102 \dots + 0.76316 \dots}{- 2 \cdot 1.01}

Multiply the numbers:

2 \cdot 1.01 = 2.02

= \frac{0.18102 \dots + 0.76316 \dots}{- 2.02}

Apply the fraction rule:

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

= - \frac{0.18102 \dots + 0.76316 \dots}{2.02}

u = \frac{- ( - 0.18102 \dots ) - 0.76316 \dots}{2 ( - 1.01 )} : \frac{0.76316 \dots - 0.18102 \dots}{2.02}

\frac{- ( - 0.18102 \dots ) - 0.76316 \dots}{2 ( - 1.01 )}

Remove parentheses:

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

= \frac{0.18102 \dots - 0.76316 \dots}{- 2 \cdot 1.01}

Multiply the numbers:

2 \cdot 1.01 = 2.02

= \frac{0.18102 \dots - 0.76316 \dots}{- 2.02}

Apply the fraction rule:

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

0.18102 \dots - 0.76316 \dots = - (0.76316 \dots - 0.18102 \dots)

= \frac{0.76316 \dots - 0.18102 \dots}{2.02}

The solutions to the quadratic equation are:

u = - \frac{0.18102 \dots + 0.76316 \dots}{2.02}, u = \frac{0.76316 \dots - 0.18102 \dots}{2.02}

Substitute back

u = cos (θ)

cos (θ) = - \frac{0.18102 \dots + 0.76316 \dots}{2.02}, cos (θ) = \frac{0.76316 \dots - 0.18102 \dots}{2.02}

cos (θ) = - \frac{0.18102 \dots + 0.76316 \dots}{2.02}, cos (θ) = \frac{0.76316 \dots - 0.18102 \dots}{2.02}

cos (θ) = - \frac{0.18102 \dots + 0.76316 \dots}{2.02} : θ = arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn, θ = - arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn

cos (θ) = - \frac{0.18102 \dots + 0.76316 \dots}{2.02}

Apply trig inverse properties

cos (θ) = - \frac{0.18102 \dots + 0.76316 \dots}{2.02}

General solutions for

cos (θ) = - \frac{0.18102 \dots + 0.76316 \dots}{2.02}

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

θ = arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn, θ = - arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn

θ = arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn, θ = - arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn

cos (θ) = \frac{0.76316 \dots - 0.18102 \dots}{2.02} : θ = arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn

cos (θ) = \frac{0.76316 \dots - 0.18102 \dots}{2.02}

Apply trig inverse properties

cos (θ) = \frac{0.76316 \dots - 0.18102 \dots}{2.02}

General solutions for

cos (θ) = \frac{0.76316 \dots - 0.18102 \dots}{2.02}

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

θ = arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn

θ = arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn

Combine all the solutions

θ = arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn, θ = - arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn, θ = arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sin (θ) - 0.1 cos (θ) = \frac{8.87}{9.8}

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

Check the solution

arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn :

True

arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn

Plug in

n = 1

arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 π 1

For

sin (θ) - 0.1 cos (θ) = \frac{8.87}{9.8}

plug in

θ = arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 π 1

sin (arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 π 1) - 0.1 cos (arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 π 1) = \frac{8.87}{9.8}

Refine

0.90510 \dots = 0.90510 \dots

\Rightarrow True

Check the solution

- arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn :

False

- arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn

Plug in

n = 1

- arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 π 1

For

sin (θ) - 0.1 cos (θ) = \frac{8.87}{9.8}

plug in

θ = - arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 π 1

sin (- arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 π 1) - 0.1 cos (- arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 π 1) = \frac{8.87}{9.8}

Refine

- 0.80068 \dots = 0.90510 \dots

\Rightarrow False

Check the solution

arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn :

True

arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn

Plug in

n = 1

arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 π 1

For

sin (θ) - 0.1 cos (θ) = \frac{8.87}{9.8}

plug in

θ = arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 π 1

sin (arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 π 1) - 0.1 cos (arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 π 1) = \frac{8.87}{9.8}

Refine

0.90510 \dots = 0.90510 \dots

\Rightarrow True

Check the solution

2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn :

False

2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 π 1

For

sin (θ) - 0.1 cos (θ) = \frac{8.87}{9.8}

plug in

θ = 2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 π 1

sin (2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 π 1) - 0.1 cos (2 π - arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 π 1) = \frac{8.87}{9.8}

Refine

- 0.97367 \dots = 0.90510 \dots

\Rightarrow False

θ = arccos (- \frac{0.18102 \dots + 0.76316 \dots}{2.02}) + 2 πn, θ = arccos (\frac{0.76316 \dots - 0.18102 \dots}{2.02}) + 2 πn

Show solutions in decimal form

θ = 2.12008 \dots + 2 πn, θ = 1.22084 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)-0.1cos(θ)=(8.87)/(9.8) ?
The general solution for sin(θ)-0.1cos(θ)=(8.87)/(9.8) is θ=2.12008…+2pin,θ=1.22084…+2pin