What is the general solution for sin(A)-0.6*cos(A)=(2.77)/(9.8) ?

The general solution for sin(A)-0.6*cos(A)=(2.77)/(9.8) is A=-2.84598…+2pin,A=0.78523…+2pin

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Popular Trigonometry >

sin(A)-0.6*cos(A)=(2.77)/(9.8)

Solution

sin (A) - 0.6 \cdot cos (A) = \frac{2.77}{9.8}

Solution

A = - 2.84598 \dots + 2 πn, A = 0.78523 \dots + 2 πn

Degrees

A = - 163.06288 \dots^{\circ} + 36 0^{\circ} n, A = 44.99039 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

Add

0.6 cos (A)

to both sides

sin (A) = 0.28265 \dots + 0.6 cos (A)

Square both sides

sin^{2} (A) = (0.28265 \dots + 0.6 cos (A))^{2}

Subtract

(0.28265 \dots + 0.6 cos (A))^{2}

from both sides

sin^{2} (A) - 0.07989 \dots - 0.33918 \dots cos (A) - 0.36 cos^{2} (A) = 0

Rewrite using trig identities

- 0.07989 \dots + sin^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A)

Use the Pythagorean identity:

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

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

= - 0.07989 \dots + 1 - cos^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A)

Simplify

- 0.07989 \dots + 1 - cos^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A) : - 1.36 cos^{2} (A) - 0.33918 \dots cos (A) + 0.92010 \dots

- 0.07989 \dots + 1 - cos^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A)

Group like terms

= - cos^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A) - 0.07989 \dots + 1

Add similar elements:

- cos^{2} (A) - 0.36 cos^{2} (A) = - 1.36 cos^{2} (A)

= - 1.36 cos^{2} (A) - 0.33918 \dots cos (A) - 0.07989 \dots + 1

Add/Subtract the numbers:

- 0.07989 \dots + 1 = 0.92010 \dots

= - 1.36 cos^{2} (A) - 0.33918 \dots cos (A) + 0.92010 \dots

= - 1.36 cos^{2} (A) - 0.33918 \dots cos (A) + 0.92010 \dots

0.92010 \dots - 0.33918 \dots cos (A) - 1.36 cos^{2} (A) = 0

Solve by substitution

0.92010 \dots - 0.33918 \dots cos (A) - 1.36 cos^{2} (A) = 0

Let:

cos (A) = u

0.92010 \dots - 0.33918 \dots u - 1.36 u^{2} = 0

0.92010 \dots - 0.33918 \dots u - 1.36 u^{2} = 0 : u = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}, u = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

0.92010 \dots - 0.33918 \dots u - 1.36 u^{2} = 0

Write in the standard form

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

- 1.36 u^{2} - 0.33918 \dots u + 0.92010 \dots = 0

Solve with the quadratic formula

- 1.36 u^{2} - 0.33918 \dots u + 0.92010 \dots = 0

Quadratic Equation Formula:

For

a = - 1.36, b = - 0.33918 \dots, c = 0.92010 \dots

u_{1, 2} = \frac{- ( - 0.33918 \dots ) \pm ( - 0.33918 \dots ) ^{2} - 4 ( - 1.36 ) \cdot 0.92010 \dots}{2 ( - 1.36 )}

u_{1, 2} = \frac{- ( - 0.33918 \dots ) \pm ( - 0.33918 \dots ) ^{2} - 4 ( - 1.36 ) \cdot 0.92010 \dots}{2 ( - 1.36 )}

(- 0.33918 \dots)^{2} - 4 (- 1.36) \cdot 0.92010 \dots = 5.12042 \dots

(- 0.33918 \dots)^{2} - 4 (- 1.36) \cdot 0.92010 \dots

Apply rule

- (- a) = a

= (- 0.33918 \dots)^{2} + 4 \cdot 1.36 \cdot 0.92010 \dots

Apply exponent rule:

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

n

is even

(- 0.33918 \dots)^{2} = 0.33918 \dots^{2}

= 0.33918 \dots^{2} + 4 \cdot 0.92010 \dots \cdot 1.36

Multiply the numbers:

4 \cdot 1.36 \cdot 0.92010 \dots = 5.00538 \dots

= 0.33918 \dots^{2} + 5.00538 \dots

0.33918 \dots^{2} = 0.11504 \dots

= 0.11504 \dots + 5.00538 \dots

Add the numbers:

0.11504 \dots + 5.00538 \dots = 5.12042 \dots

= 5.12042 \dots

u_{1, 2} = \frac{- ( - 0.33918 \dots ) \pm 5.12042 \dots}{2 ( - 1.36 )}

Separate the solutions

u_{1} = \frac{- ( - 0.33918 \dots ) + 5.12042 \dots}{2 ( - 1.36 )}, u_{2} = \frac{- ( - 0.33918 \dots ) - 5.12042 \dots}{2 ( - 1.36 )}

u = \frac{- ( - 0.33918 \dots ) + 5.12042 \dots}{2 ( - 1.36 )} : - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

\frac{- ( - 0.33918 \dots ) + 5.12042 \dots}{2 ( - 1.36 )}

Remove parentheses:

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

= \frac{0.33918 \dots + 5.12042 \dots}{- 2 \cdot 1.36}

Multiply the numbers:

2 \cdot 1.36 = 2.72

= \frac{0.33918 \dots + 5.12042 \dots}{- 2.72}

Apply the fraction rule:

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

= - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

u = \frac{- ( - 0.33918 \dots ) - 5.12042 \dots}{2 ( - 1.36 )} : \frac{5.12042 \dots - 0.33918 \dots}{2.72}

\frac{- ( - 0.33918 \dots ) - 5.12042 \dots}{2 ( - 1.36 )}

Remove parentheses:

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

= \frac{0.33918 \dots - 5.12042 \dots}{- 2 \cdot 1.36}

Multiply the numbers:

2 \cdot 1.36 = 2.72

= \frac{0.33918 \dots - 5.12042 \dots}{- 2.72}

Apply the fraction rule:

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

0.33918 \dots - 5.12042 \dots = - (5.12042 \dots - 0.33918 \dots)

= \frac{5.12042 \dots - 0.33918 \dots}{2.72}

The solutions to the quadratic equation are:

u = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}, u = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

Substitute back

u = cos (A)

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}, cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}, cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72} : A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

Apply trig inverse properties

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

General solutions for

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

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

A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72} : A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn, A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

Apply trig inverse properties

cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

General solutions for

cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

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

A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn, A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn, A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

Combine all the solutions

A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn, A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

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

Check the solution

arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn :

False

arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

Plug in

n = 1

arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1

For

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

plug in

A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1

sin (arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1) - 0.6 cos (arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1) = \frac{2.77}{9.8}

Refine

0.86529 \dots = 0.28265 \dots

\Rightarrow False

Check the solution

- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn :

True

- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

Plug in

n = 1

- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1

For

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

plug in

A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1

sin (- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1) - 0.6 cos (- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1) = \frac{2.77}{9.8}

Refine

0.28265 \dots = 0.28265 \dots

\Rightarrow True

Check the solution

arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn :

True

arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

Plug in

n = 1

arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1

For

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

plug in

A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1

sin (arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1) - 0.6 cos (arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1) = \frac{2.77}{9.8}

Refine

0.28265 \dots = 0.28265 \dots

\Rightarrow True

Check the solution

2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn :

False

2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1

For

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

plug in

A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1

sin (2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1) - 0.6 cos (2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1) = \frac{2.77}{9.8}

Refine

- 1.13132 \dots = 0.28265 \dots

\Rightarrow False

A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

Show solutions in decimal form

A = - 2.84598 \dots + 2 πn, A = 0.78523 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(A)-0.6*cos(A)=(2.77)/(9.8) ?
The general solution for sin(A)-0.6*cos(A)=(2.77)/(9.8) is A=-2.84598…+2pin,A=0.78523…+2pin