What is the general solution for 12*9.8*sin(a)-0.6*12*9.8*cos(a)=12*1.79 ?

The general solution for 12*9.8*sin(a)-0.6*12*9.8*cos(a)=12*1.79 is a=-2.75844…+2pin,a=0.69769…+2pin

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Popular Trigonometry >

129.8sin(a)-0.6129.8cos(a)=121.79

Solution

12 \cdot 9.8 \cdot sin (a) - 0.6 \cdot 12 \cdot 9.8 \cdot cos (a) = 12 \cdot 1.79

Solution

a = - 2.75844 \dots + 2 πn, a = 0.69769 \dots + 2 πn

Degrees

a = - 158.04722 \dots^{\circ} + 36 0^{\circ} n, a = 39.97473 \dots^{\circ} + 36 0^{\circ} n

Solution steps

12 \cdot 9.8 sin (a) - 0.6 \cdot 12 \cdot 9.8 cos (a) = 12 \cdot 1.79

Add

0.6129.8 cos (a)

to both sides

117.6 sin (a) = 21.48 + 70.56 cos (a)

Square both sides

(117.6 sin (a))^{2} = (21.48 + 70.56 cos (a))^{2}

Subtract

(21.48 + 70.56 cos (a))^{2}

from both sides

13829.76 sin^{2} (a) - 461.3904 - 3031.2576 cos (a) - 4978.7136 cos^{2} (a) = 0

Rewrite using trig identities

- 461.3904 + 13829.76 sin^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

Use the Pythagorean identity:

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

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

= - 461.3904 + 13829.76 (1 - cos^{2} (a)) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

Simplify

- 461.3904 + 13829.76 (1 - cos^{2} (a)) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a) : - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

- 461.3904 + 13829.76 (1 - cos^{2} (a)) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

Expand

13829.76 (1 - cos^{2} (a)) : 13829.76 - 13829.76 cos^{2} (a)

13829.76 (1 - cos^{2} (a))

Apply the distributive law:

a (b - c) = ab - a c

a = 13829.76, b = 1, c = cos^{2} (a)

= 13829.76 \cdot 1 - 13829.76 cos^{2} (a)

= 1 \cdot 13829.76 - 13829.76 cos^{2} (a)

Multiply the numbers:

1 \cdot 13829.76 = 13829.76

= 13829.76 - 13829.76 cos^{2} (a)

= - 461.3904 + 13829.76 - 13829.76 cos^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

Simplify

- 461.3904 + 13829.76 - 13829.76 cos^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a) : - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

- 461.3904 + 13829.76 - 13829.76 cos^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

Group like terms

= - 13829.76 cos^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a) - 461.3904 + 13829.76

Add similar elements:

- 13829.76 cos^{2} (a) - 4978.7136 cos^{2} (a) = - 18808.4736 cos^{2} (a)

= - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) - 461.3904 + 13829.76

Add/Subtract the numbers:

- 461.3904 + 13829.76 = 13368.3696

= - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

= - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

= - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

13368.3696 - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) = 0

Solve by substitution

13368.3696 - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) = 0

Let:

cos (a) = u

13368.3696 - 18808.4736 u^{2} - 3031.2576 u = 0

13368.3696 - 18808.4736 u^{2} - 3031.2576 u = 0 : u = - \frac{3031.2576 + 1014943029.424128}{37616.9472}, u = \frac{1014943029.424128 - 3031.2576}{37616.9472}

13368.3696 - 18808.4736 u^{2} - 3031.2576 u = 0

Write in the standard form

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

- 18808.4736 u^{2} - 3031.2576 u + 13368.3696 = 0

Solve with the quadratic formula

- 18808.4736 u^{2} - 3031.2576 u + 13368.3696 = 0

Quadratic Equation Formula:

For

a = - 18808.4736, b = - 3031.2576, c = 13368.3696

u_{1, 2} = \frac{- ( - 3031.2576 ) \pm ( - 3031.2576 ) ^{2} - 4 ( - 18808.4736 ) \cdot 13368.3696}{2 ( - 18808.4736 )}

u_{1, 2} = \frac{- ( - 3031.2576 ) \pm ( - 3031.2576 ) ^{2} - 4 ( - 18808.4736 ) \cdot 13368.3696}{2 ( - 18808.4736 )}

(- 3031.2576)^{2} - 4 (- 18808.4736) \cdot 13368.3696 = 1014943029.424128

(- 3031.2576)^{2} - 4 (- 18808.4736) \cdot 13368.3696

Apply rule

- (- a) = a

= (- 3031.2576)^{2} + 4 \cdot 18808.4736 \cdot 13368.3696

Apply exponent rule:

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

n

is even

(- 3031.2576)^{2} = 3031.257 6^{2}

= 3031.257 6^{2} + 4 \cdot 13368.3696 \cdot 18808.4736

Multiply the numbers:

4 \cdot 18808.4736 \cdot 13368.3696 = 1005754506.78657 \dots

= 3031.257 6^{2} + 1005754506.78657 \dots

3031.257 6^{2} = 9188522.63755 \dots

= 9188522.63755 \dots + 1005754506.78657 \dots

Add the numbers:

9188522.63755 \dots + 1005754506.78657 \dots = 1014943029.424128

= 1014943029.424128

u_{1, 2} = \frac{- ( - 3031.2576 ) \pm 1014943029.424128}{2 ( - 18808.4736 )}

Separate the solutions

u_{1} = \frac{- ( - 3031.2576 ) + 1014943029.424128}{2 ( - 18808.4736 )}, u_{2} = \frac{- ( - 3031.2576 ) - 1014943029.424128}{2 ( - 18808.4736 )}

u = \frac{- ( - 3031.2576 ) + 1014943029.424128}{2 ( - 18808.4736 )} : - \frac{3031.2576 + 1014943029.424128}{37616.9472}

\frac{- ( - 3031.2576 ) + 1014943029.424128}{2 ( - 18808.4736 )}

Remove parentheses:

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

= \frac{3031.2576 + 1014943029.424128}{- 2 \cdot 18808.4736}

Multiply the numbers:

2 \cdot 18808.4736 = 37616.9472

= \frac{3031.2576 + 1014943029.424128}{- 37616.9472}

Apply the fraction rule:

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

= - \frac{3031.2576 + 1014943029.424128}{37616.9472}

u = \frac{- ( - 3031.2576 ) - 1014943029.424128}{2 ( - 18808.4736 )} : \frac{1014943029.424128 - 3031.2576}{37616.9472}

\frac{- ( - 3031.2576 ) - 1014943029.424128}{2 ( - 18808.4736 )}

Remove parentheses:

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

= \frac{3031.2576 - 1014943029.424128}{- 2 \cdot 18808.4736}

Multiply the numbers:

2 \cdot 18808.4736 = 37616.9472

= \frac{3031.2576 - 1014943029.424128}{- 37616.9472}

Apply the fraction rule:

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

3031.2576 - 1014943029.424128 = - (1014943029.424128 - 3031.2576)

= \frac{1014943029.424128 - 3031.2576}{37616.9472}

The solutions to the quadratic equation are:

u = - \frac{3031.2576 + 1014943029.424128}{37616.9472}, u = \frac{1014943029.424128 - 3031.2576}{37616.9472}

Substitute back

u = cos (a)

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}, cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}, cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472} : a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}

Apply trig inverse properties

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}

General solutions for

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}

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

a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472} : a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn, a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

Apply trig inverse properties

cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

General solutions for

cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

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

a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn, a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn, a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

Combine all the solutions

a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn, a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

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

Check the solution

arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn :

False

arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

Plug in

n = 1

arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1

For

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

plug in

a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1

12 \cdot 9.8 sin (arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1) - 0.6 \cdot 12 \cdot 9.8 cos (arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1) = 12 \cdot 1.79

Refine

109.40771 \dots = 21.48

\Rightarrow False

Check the solution

- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn :

True

- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

Plug in

n = 1

- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1

For

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

plug in

a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1

12 \cdot 9.8 sin (- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1) - 0.6 \cdot 12 \cdot 9.8 cos (- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1) = 12 \cdot 1.79

Refine

21.48 = 21.48

\Rightarrow True

Check the solution

arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn :

True

arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

Plug in

n = 1

arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1

For

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

plug in

a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1

12 \cdot 9.8 sin (arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1) - 0.6 \cdot 12 \cdot 9.8 cos (arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1) = 12 \cdot 1.79

Refine

21.48 = 21.48

\Rightarrow True

Check the solution

2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn :

False

2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1

For

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

plug in

a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1

12 \cdot 9.8 sin (2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1) - 0.6 \cdot 12 \cdot 9.8 cos (2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1) = 12 \cdot 1.79

Refine

- 129.62418 \dots = 21.48

\Rightarrow False

a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

Show solutions in decimal form

a = - 2.75844 \dots + 2 πn, a = 0.69769 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 12*9.8*sin(a)-0.6*12*9.8*cos(a)=12*1.79 ?
The general solution for 12*9.8*sin(a)-0.6*12*9.8*cos(a)=12*1.79 is a=-2.75844…+2pin,a=0.69769…+2pin