What is the general solution for 21.16=19.6sin(x)-29.4cos(x) ?

The general solution for 21.16=19.6sin(x)-29.4cos(x) is x=-2.80086…+2pin,x=1.62485…+2pin

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Popular Trigonometry >

21.16=19.6sin(x)-29.4cos(x)

Solution

21.16 = 19.6 sin (x) - 29.4 cos (x)

Solution

x = - 2.80086 \dots + 2 πn, x = 1.62485 \dots + 2 πn

Degrees

x = - 160.47763 \dots^{\circ} + 36 0^{\circ} n, x = 93.09749 \dots^{\circ} + 36 0^{\circ} n

Solution steps

21.16 = 19.6 sin (x) - 29.4 cos (x)

Add

29.4 cos (x)

to both sides

19.6 sin (x) = 21.16 + 29.4 cos (x)

Square both sides

(19.6 sin (x))^{2} = (21.16 + 29.4 cos (x))^{2}

Subtract

(21.16 + 29.4 cos (x))^{2}

from both sides

384.16 sin^{2} (x) - 447.7456 - 1244.208 cos (x) - 864.36 cos^{2} (x) = 0

Rewrite using trig identities

- 447.7456 - 1244.208 cos (x) + 384.16 sin^{2} (x) - 864.36 cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x)

Simplify

- 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x) : - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x)

Expand

384.16 (1 - cos^{2} (x)) : 384.16 - 384.16 cos^{2} (x)

384.16 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 384.16, b = 1, c = cos^{2} (x)

= 384.16 \cdot 1 - 384.16 cos^{2} (x)

= 1 \cdot 384.16 - 384.16 cos^{2} (x)

Multiply the numbers:

1 \cdot 384.16 = 384.16

= 384.16 - 384.16 cos^{2} (x)

= - 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x)

Simplify

- 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x) : - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x)

Add similar elements:

- 384.16 cos^{2} (x) - 864.36 cos^{2} (x) = - 1248.52 cos^{2} (x)

= - 447.7456 - 1244.208 cos (x) + 384.16 - 1248.52 cos^{2} (x)

Group like terms

= - 1244.208 cos (x) - 1248.52 cos^{2} (x) - 447.7456 + 384.16

Add/Subtract the numbers:

- 447.7456 + 384.16 = - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 63.5856 - 1244.208 cos (x) - 1248.52 cos^{2} (x) = 0

Solve by substitution

- 63.5856 - 1244.208 cos (x) - 1248.52 cos^{2} (x) = 0

Let:

cos (x) = u

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0 : u = - \frac{1244.208 + 1230501.974016}{2497.04}, u = - \frac{1244.208 - 1230501.974016}{2497.04}

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0

Write in the standard form

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

- 1248.52 u^{2} - 1244.208 u - 63.5856 = 0

Solve with the quadratic formula

- 1248.52 u^{2} - 1244.208 u - 63.5856 = 0

Quadratic Equation Formula:

For

a = - 1248.52, b = - 1244.208, c = - 63.5856

u_{1, 2} = \frac{- ( - 1244.208 ) \pm ( - 1244.208 ) ^{2} - 4 ( - 1248.52 ) ( - 63.5856 )}{2 ( - 1248.52 )}

u_{1, 2} = \frac{- ( - 1244.208 ) \pm ( - 1244.208 ) ^{2} - 4 ( - 1248.52 ) ( - 63.5856 )}{2 ( - 1248.52 )}

(- 1244.208)^{2} - 4 (- 1248.52) (- 63.5856) = 1230501.974016

(- 1244.208)^{2} - 4 (- 1248.52) (- 63.5856)

Apply rule

- (- a) = a

= (- 1244.208)^{2} - 4 \cdot 1248.52 \cdot 63.5856

Apply exponent rule:

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

n

is even

(- 1244.208)^{2} = 1244.20 8^{2}

= 1244.20 8^{2} - 4 \cdot 63.5856 \cdot 1248.52

Multiply the numbers:

4 \cdot 1248.52 \cdot 63.5856 = 317551.573248

= 1244.20 8^{2} - 317551.573248

1244.20 8^{2} = 1548053.547264

= 1548053.547264 - 317551.573248

Subtract the numbers:

1548053.547264 - 317551.573248 = 1230501.974016

= 1230501.974016

u_{1, 2} = \frac{- ( - 1244.208 ) \pm 1230501.974016}{2 ( - 1248.52 )}

Separate the solutions

u_{1} = \frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )}, u_{2} = \frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )}

u = \frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )} : - \frac{1244.208 + 1230501.974016}{2497.04}

\frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )}

Remove parentheses:

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

= \frac{1244.208 + 1230501.974016}{- 2 \cdot 1248.52}

Multiply the numbers:

2 \cdot 1248.52 = 2497.04

= \frac{1244.208 + 1230501.974016}{- 2497.04}

Apply the fraction rule:

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

= - \frac{1244.208 + 1230501.974016}{2497.04}

u = \frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )} : - \frac{1244.208 - 1230501.974016}{2497.04}

\frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )}

Remove parentheses:

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

= \frac{1244.208 - 1230501.974016}{- 2 \cdot 1248.52}

Multiply the numbers:

2 \cdot 1248.52 = 2497.04

= \frac{1244.208 - 1230501.974016}{- 2497.04}

Apply the fraction rule:

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

= - \frac{1244.208 - 1230501.974016}{2497.04}

The solutions to the quadratic equation are:

u = - \frac{1244.208 + 1230501.974016}{2497.04}, u = - \frac{1244.208 - 1230501.974016}{2497.04}

Substitute back

u = cos (x)

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}, cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}, cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04} : x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

Apply trig inverse properties

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

General solutions for

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

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

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04} : x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

Apply trig inverse properties

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

General solutions for

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

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

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

Combine all the solutions

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

19.6 sin (x) - 29.4 cos (x) = 21.16

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

Check the solution

arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn :

False

arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

Plug in

n = 1

arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

For

19.6 sin (x) - 29.4 cos (x) = 21.16

plug in

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) = 21.16

Refine

34.25965 \dots = 21.16

\Rightarrow False

Check the solution

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn :

True

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

Plug in

n = 1

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

For

19.6 sin (x) - 29.4 cos (x) = 21.16

plug in

x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) = 21.16

Refine

21.16 = 21.16

\Rightarrow True

Check the solution

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn :

True

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

Plug in

n = 1

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

For

19.6 sin (x) - 29.4 cos (x) = 21.16

plug in

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) = 21.16

Refine

21.16 = 21.16

\Rightarrow True

Check the solution

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn :

False

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

Plug in

n = 1

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

For

19.6 sin (x) - 29.4 cos (x) = 21.16

plug in

x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) = 21.16

Refine

- 17.98273 \dots = 21.16

\Rightarrow False

x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

Show solutions in decimal form

x = - 2.80086 \dots + 2 πn, x = 1.62485 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 21.16=19.6sin(x)-29.4cos(x) ?
The general solution for 21.16=19.6sin(x)-29.4cos(x) is x=-2.80086…+2pin,x=1.62485…+2pin