What is the general solution for 9.8*sin(α)-1.96*cos(α)=0.47 ?

The general solution for 9.8*sin(α)-1.96*cos(α)=0.47 is α=-2.99124…+2pin,α=0.24444…+2pin

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Popular Trigonometry >

9.8sin(α)-1.96cos(α)=0.47

Solution

9.8 \cdot sin (α) - 1.96 \cdot cos (α) = 0.47

Solution

α = - 2.99124 \dots + 2 πn, α = 0.24444 \dots + 2 πn

Degrees

α = - 171.38555 \dots^{\circ} + 36 0^{\circ} n, α = 14.00542 \dots^{\circ} + 36 0^{\circ} n

Solution steps

9.8 sin (α) - 1.96 cos (α) = 0.47

Add

1.96 cos (α)

to both sides

9.8 sin (α) = 0.47 + 1.96 cos (α)

Square both sides

(9.8 sin (α))^{2} = (0.47 + 1.96 cos (α))^{2}

Subtract

(0.47 + 1.96 cos (α))^{2}

from both sides

96.04 sin^{2} (α) - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) = 0

Rewrite using trig identities

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 sin^{2} (α)

Use the Pythagorean identity:

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

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

= - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

Simplify

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α)) : - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

Expand

96.04 (1 - cos^{2} (α)) : 96.04 - 96.04 cos^{2} (α)

96.04 (1 - cos^{2} (α))

Apply the distributive law:

a (b - c) = ab - a c

a = 96.04, b = 1, c = cos^{2} (α)

= 96.04 \cdot 1 - 96.04 cos^{2} (α)

= 1 \cdot 96.04 - 96.04 cos^{2} (α)

Multiply the numbers:

1 \cdot 96.04 = 96.04

= 96.04 - 96.04 cos^{2} (α)

= - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

Simplify

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α) : - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

Group like terms

= - 1.8424 cos (α) - 3.8416 cos^{2} (α) - 96.04 cos^{2} (α) - 0.2209 + 96.04

Add similar elements:

- 3.8416 cos^{2} (α) - 96.04 cos^{2} (α) = - 99.8816 cos^{2} (α)

= - 1.8424 cos (α) - 99.8816 cos^{2} (α) - 0.2209 + 96.04

Add/Subtract the numbers:

- 0.2209 + 96.04 = 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

95.8191 - 1.8424 cos (α) - 99.8816 cos^{2} (α) = 0

Solve by substitution

95.8191 - 1.8424 cos (α) - 99.8816 cos^{2} (α) = 0

Let:

cos (α) = u

95.8191 - 1.8424 u - 99.8816 u^{2} = 0

95.8191 - 1.8424 u - 99.8816 u^{2} = 0 : u = - \frac{1.8424 + 38285.654512}{199.7632}, u = \frac{38285.654512 - 1.8424}{199.7632}

95.8191 - 1.8424 u - 99.8816 u^{2} = 0

Write in the standard form

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

- 99.8816 u^{2} - 1.8424 u + 95.8191 = 0

Solve with the quadratic formula

- 99.8816 u^{2} - 1.8424 u + 95.8191 = 0

Quadratic Equation Formula:

For

a = - 99.8816, b = - 1.8424, c = 95.8191

u_{1, 2} = \frac{- ( - 1.8424 ) \pm ( - 1.8424 ) ^{2} - 4 ( - 99.8816 ) \cdot 95.8191}{2 ( - 99.8816 )}

u_{1, 2} = \frac{- ( - 1.8424 ) \pm ( - 1.8424 ) ^{2} - 4 ( - 99.8816 ) \cdot 95.8191}{2 ( - 99.8816 )}

(- 1.8424)^{2} - 4 (- 99.8816) \cdot 95.8191 = 38285.654512

(- 1.8424)^{2} - 4 (- 99.8816) \cdot 95.8191

Apply rule

- (- a) = a

= (- 1.8424)^{2} + 4 \cdot 99.8816 \cdot 95.8191

Apply exponent rule:

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

n

is even

(- 1.8424)^{2} = 1.842 4^{2}

= 1.842 4^{2} + 4 \cdot 95.8191 \cdot 99.8816

Multiply the numbers:

4 \cdot 99.8816 \cdot 95.8191 = 38282.26007 \dots

= 1.842 4^{2} + 38282.26007 \dots

1.842 4^{2} = 3.39443776

= 3.39443776 + 38282.26007 \dots

Add the numbers:

3.39443776 + 38282.26007 \dots = 38285.654512

= 38285.654512

u_{1, 2} = \frac{- ( - 1.8424 ) \pm 38285.654512}{2 ( - 99.8816 )}

Separate the solutions

u_{1} = \frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )}, u_{2} = \frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )}

u = \frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )} : - \frac{1.8424 + 38285.654512}{199.7632}

\frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )}

Remove parentheses:

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

= \frac{1.8424 + 38285.654512}{- 2 \cdot 99.8816}

Multiply the numbers:

2 \cdot 99.8816 = 199.7632

= \frac{1.8424 + 38285.654512}{- 199.7632}

Apply the fraction rule:

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

= - \frac{1.8424 + 38285.654512}{199.7632}

u = \frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )} : \frac{38285.654512 - 1.8424}{199.7632}

\frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )}

Remove parentheses:

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

= \frac{1.8424 - 38285.654512}{- 2 \cdot 99.8816}

Multiply the numbers:

2 \cdot 99.8816 = 199.7632

= \frac{1.8424 - 38285.654512}{- 199.7632}

Apply the fraction rule:

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

1.8424 - 38285.654512 = - (38285.654512 - 1.8424)

= \frac{38285.654512 - 1.8424}{199.7632}

The solutions to the quadratic equation are:

u = - \frac{1.8424 + 38285.654512}{199.7632}, u = \frac{38285.654512 - 1.8424}{199.7632}

Substitute back

u = cos (α)

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}, cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}, cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632} : α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

Apply trig inverse properties

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

General solutions for

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

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

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

cos (α) = \frac{38285.654512 - 1.8424}{199.7632} : α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

Apply trig inverse properties

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

General solutions for

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

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

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

Combine all the solutions

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

9.8 sin (α) - 1.96 cos (α) = 0.47

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

Check the solution

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn :

False

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

Plug in

n = 1

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

For

9.8 sin (α) - 1.96 cos (α) = 0.47

plug in

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

9.8 sin (arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) - 1.96 cos (arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) = 0.47

Refine

3.40577 \dots = 0.47

\Rightarrow False

Check the solution

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn :

True

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

Plug in

n = 1

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

For

9.8 sin (α) - 1.96 cos (α) = 0.47

plug in

α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

9.8 sin (- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) - 1.96 cos (- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) = 0.47

Refine

0.47 = 0.47

\Rightarrow True

Check the solution

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn :

True

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

Plug in

n = 1

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

For

9.8 sin (α) - 1.96 cos (α) = 0.47

plug in

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

9.8 sin (arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) - 1.96 cos (arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) = 0.47

Refine

0.47 = 0.47

\Rightarrow True

Check the solution

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn :

False

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

For

9.8 sin (α) - 1.96 cos (α) = 0.47

plug in

α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

9.8 sin (2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) - 1.96 cos (2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) = 0.47

Refine

- 4.27346 \dots = 0.47

\Rightarrow False

α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

Show solutions in decimal form

α = - 2.99124 \dots + 2 πn, α = 0.24444 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 9.8*sin(α)-1.96*cos(α)=0.47 ?
The general solution for 9.8*sin(α)-1.96*cos(α)=0.47 is α=-2.99124…+2pin,α=0.24444…+2pin