What is the general solution for 196sin(θ)-49cos(θ)=160 ?

The general solution for 196sin(θ)-49cos(θ)=160 is θ=2.47257…+2pin,θ=1.15897…+2pin

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Popular Trigonometry >

196sin(θ)-49cos(θ)=160

Solution

196 sin (θ) - 49 cos (θ) = 160

Solution

θ = 2.47257 \dots + 2 πn, θ = 1.15897 \dots + 2 πn

Degrees

θ = 141.66783 \dots^{\circ} + 36 0^{\circ} n, θ = 66.40464 \dots^{\circ} + 36 0^{\circ} n

Solution steps

196 sin (θ) - 49 cos (θ) = 160

Add

49 cos (θ)

to both sides

196 sin (θ) = 160 + 49 cos (θ)

Square both sides

(196 sin (θ))^{2} = (160 + 49 cos (θ))^{2}

Subtract

(160 + 49 cos (θ))^{2}

from both sides

38416 sin^{2} (θ) - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) = 0

Rewrite using trig identities

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 sin^{2} (θ)

Use the Pythagorean identity:

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

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

= - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ))

Simplify

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ)) : - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ))

Expand

38416 (1 - cos^{2} (θ)) : 38416 - 38416 cos^{2} (θ)

38416 (1 - cos^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 38416, b = 1, c = cos^{2} (θ)

= 38416 \cdot 1 - 38416 cos^{2} (θ)

Multiply the numbers:

38416 \cdot 1 = 38416

= 38416 - 38416 cos^{2} (θ)

= - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ)

Simplify

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ) : - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ)

Group like terms

= - 15680 cos (θ) - 2401 cos^{2} (θ) - 38416 cos^{2} (θ) - 25600 + 38416

Add similar elements:

- 2401 cos^{2} (θ) - 38416 cos^{2} (θ) = - 40817 cos^{2} (θ)

= - 15680 cos (θ) - 40817 cos^{2} (θ) - 25600 + 38416

Add/Subtract the numbers:

- 25600 + 38416 = 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

12816 - 15680 cos (θ) - 40817 cos^{2} (θ) = 0

Solve by substitution

12816 - 15680 cos (θ) - 40817 cos^{2} (θ) = 0

Let:

cos (θ) = u

12816 - 15680 u - 40817 u^{2} = 0

12816 - 15680 u - 40817 u^{2} = 0 : u = - \frac{15680 + 2338305088}{81634}, u = \frac{2338305088 - 15680}{81634}

12816 - 15680 u - 40817 u^{2} = 0

Write in the standard form

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

- 40817 u^{2} - 15680 u + 12816 = 0

Solve with the quadratic formula

- 40817 u^{2} - 15680 u + 12816 = 0

Quadratic Equation Formula:

For

a = - 40817, b = - 15680, c = 12816

u_{1, 2} = \frac{- ( - 15680 ) \pm ( - 15680 ) ^{2} - 4 ( - 40817 ) \cdot 12816}{2 ( - 40817 )}

u_{1, 2} = \frac{- ( - 15680 ) \pm ( - 15680 ) ^{2} - 4 ( - 40817 ) \cdot 12816}{2 ( - 40817 )}

(- 15680)^{2} - 4 (- 40817) \cdot 12816 = 2338305088

(- 15680)^{2} - 4 (- 40817) \cdot 12816

Apply rule

- (- a) = a

= (- 15680)^{2} + 4 \cdot 40817 \cdot 12816

Apply exponent rule:

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

n

is even

(- 15680)^{2} = 1568 0^{2}

= 1568 0^{2} + 4 \cdot 40817 \cdot 12816

Multiply the numbers:

4 \cdot 40817 \cdot 12816 = 2092442688

= 1568 0^{2} + 2092442688

1568 0^{2} = 245862400

= 245862400 + 2092442688

Add the numbers:

245862400 + 2092442688 = 2338305088

= 2338305088

u_{1, 2} = \frac{- ( - 15680 ) \pm 2338305088}{2 ( - 40817 )}

Separate the solutions

u_{1} = \frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )}, u_{2} = \frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )}

u = \frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )} : - \frac{15680 + 2338305088}{81634}

\frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )}

Remove parentheses:

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

= \frac{15680 + 2338305088}{- 2 \cdot 40817}

Multiply the numbers:

2 \cdot 40817 = 81634

= \frac{15680 + 2338305088}{- 81634}

Apply the fraction rule:

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

= - \frac{15680 + 2338305088}{81634}

u = \frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )} : \frac{2338305088 - 15680}{81634}

\frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )}

Remove parentheses:

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

= \frac{15680 - 2338305088}{- 2 \cdot 40817}

Multiply the numbers:

2 \cdot 40817 = 81634

= \frac{15680 - 2338305088}{- 81634}

Apply the fraction rule:

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

15680 - 2338305088 = - (2338305088 - 15680)

= \frac{2338305088 - 15680}{81634}

The solutions to the quadratic equation are:

u = - \frac{15680 + 2338305088}{81634}, u = \frac{2338305088 - 15680}{81634}

Substitute back

u = cos (θ)

cos (θ) = - \frac{15680 + 2338305088}{81634}, cos (θ) = \frac{2338305088 - 15680}{81634}

cos (θ) = - \frac{15680 + 2338305088}{81634}, cos (θ) = \frac{2338305088 - 15680}{81634}

cos (θ) = - \frac{15680 + 2338305088}{81634} : θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

cos (θ) = - \frac{15680 + 2338305088}{81634}

Apply trig inverse properties

cos (θ) = - \frac{15680 + 2338305088}{81634}

General solutions for

cos (θ) = - \frac{15680 + 2338305088}{81634}

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

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

cos (θ) = \frac{2338305088 - 15680}{81634} : θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

cos (θ) = \frac{2338305088 - 15680}{81634}

Apply trig inverse properties

cos (θ) = \frac{2338305088 - 15680}{81634}

General solutions for

cos (θ) = \frac{2338305088 - 15680}{81634}

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

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Combine all the solutions

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

196 sin (θ) - 49 cos (θ) = 160

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

Check the solution

arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn :

True

arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

Plug in

n = 1

arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

For

196 sin (θ) - 49 cos (θ) = 160

plug in

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

196 sin (arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) - 49 cos (arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) = 160

Refine

160 = 160

\Rightarrow True

Check the solution

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn :

False

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

Plug in

n = 1

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

For

196 sin (θ) - 49 cos (θ) = 160

plug in

θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

196 sin (- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) - 49 cos (- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) = 160

Refine

- 83.12602 \dots = 160

\Rightarrow False

Check the solution

arccos (\frac{2338305088 - 15680}{81634}) + 2 πn :

True

arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Plug in

n = 1

arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

For

196 sin (θ) - 49 cos (θ) = 160

plug in

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

196 sin (arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) - 49 cos (arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) = 160

Refine

160 = 160

\Rightarrow True

Check the solution

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn :

False

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

For

196 sin (θ) - 49 cos (θ) = 160

plug in

θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

196 sin (2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) - 49 cos (2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) = 160

Refine

- 199.22691 \dots = 160

\Rightarrow False

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Show solutions in decimal form

θ = 2.47257 \dots + 2 πn, θ = 1.15897 \dots + 2 πn

Graph

Popular Examples

cot(b)=(tan(b)cot(b))/(csc(b))

Frequently Asked Questions (FAQ)

What is the general solution for 196sin(θ)-49cos(θ)=160 ?
The general solution for 196sin(θ)-49cos(θ)=160 is θ=2.47257…+2pin,θ=1.15897…+2pin