What is the general solution for 49.55cos(θ)-30sin(θ)=1.225 ?

The general solution for 49.55cos(θ)-30sin(θ)=1.225 is θ=pi+1.04752…+2pin,θ=1.00522…+2pin

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Popular Trigonometry >

49.55cos(θ)-30sin(θ)=1.225

Solution

49.55 cos (θ) - 30 sin (θ) = 1.225

Solution

θ = π + 1.04752 \dots + 2 πn, θ = 1.00522 \dots + 2 πn

Degrees

θ = 240.01903 \dots^{\circ} + 36 0^{\circ} n, θ = 57.59542 \dots^{\circ} + 36 0^{\circ} n

Solution steps

49.55 cos (θ) - 30 sin (θ) = 1.225

Add

30 sin (θ)

to both sides

49.55 cos (θ) = 1.225 + 30 sin (θ)

Square both sides

(49.55 cos (θ))^{2} = (1.225 + 30 sin (θ))^{2}

Subtract

(1.225 + 30 sin (θ))^{2}

from both sides

2455.2025 cos^{2} (θ) - 1.500625 - 73.5 sin (θ) - 900 sin^{2} (θ) = 0

Rewrite using trig identities

- 1.500625 + 2455.2025 cos^{2} (θ) - 73.5 sin (θ) - 900 sin^{2} (θ)

Use the Pythagorean identity:

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

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

= - 1.500625 + 2455.2025 (1 - sin^{2} (θ)) - 73.5 sin (θ) - 900 sin^{2} (θ)

Simplify

- 1.500625 + 2455.2025 (1 - sin^{2} (θ)) - 73.5 sin (θ) - 900 sin^{2} (θ) : - 3355.2025 sin^{2} (θ) - 73.5 sin (θ) + 2453.701875

- 1.500625 + 2455.2025 (1 - sin^{2} (θ)) - 73.5 sin (θ) - 900 sin^{2} (θ)

Expand

2455.2025 (1 - sin^{2} (θ)) : 2455.2025 - 2455.2025 sin^{2} (θ)

2455.2025 (1 - sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 2455.2025, b = 1, c = sin^{2} (θ)

= 2455.2025 \cdot 1 - 2455.2025 sin^{2} (θ)

= 1 \cdot 2455.2025 - 2455.2025 sin^{2} (θ)

Multiply the numbers:

1 \cdot 2455.2025 = 2455.2025

= 2455.2025 - 2455.2025 sin^{2} (θ)

= - 1.500625 + 2455.2025 - 2455.2025 sin^{2} (θ) - 73.5 sin (θ) - 900 sin^{2} (θ)

Simplify

- 1.500625 + 2455.2025 - 2455.2025 sin^{2} (θ) - 73.5 sin (θ) - 900 sin^{2} (θ) : - 3355.2025 sin^{2} (θ) - 73.5 sin (θ) + 2453.701875

- 1.500625 + 2455.2025 - 2455.2025 sin^{2} (θ) - 73.5 sin (θ) - 900 sin^{2} (θ)

Group like terms

= - 2455.2025 sin^{2} (θ) - 73.5 sin (θ) - 900 sin^{2} (θ) - 1.500625 + 2455.2025

Add similar elements:

- 2455.2025 sin^{2} (θ) - 900 sin^{2} (θ) = - 3355.2025 sin^{2} (θ)

= - 3355.2025 sin^{2} (θ) - 73.5 sin (θ) - 1.500625 + 2455.2025

Add/Subtract the numbers:

- 1.500625 + 2455.2025 = 2453.701875

= - 3355.2025 sin^{2} (θ) - 73.5 sin (θ) + 2453.701875

= - 3355.2025 sin^{2} (θ) - 73.5 sin (θ) + 2453.701875

= - 3355.2025 sin^{2} (θ) - 73.5 sin (θ) + 2453.701875

2453.701875 - 3355.2025 sin^{2} (θ) - 73.5 sin (θ) = 0

Solve by substitution

2453.701875 - 3355.2025 sin^{2} (θ) - 73.5 sin (θ) = 0

Let:

sin (θ) = u

2453.701875 - 3355.2025 u^{2} - 73.5 u = 0

2453.701875 - 3355.2025 u^{2} - 73.5 u = 0 : u = - \frac{73.5 + 32936068.91101 \dots}{6710.405}, u = \frac{32936068.91101 \dots - 73.5}{6710.405}

2453.701875 - 3355.2025 u^{2} - 73.5 u = 0

Write in the standard form

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

- 3355.2025 u^{2} - 73.5 u + 2453.701875 = 0

Solve with the quadratic formula

- 3355.2025 u^{2} - 73.5 u + 2453.701875 = 0

Quadratic Equation Formula:

For

a = - 3355.2025, b = - 73.5, c = 2453.701875

u_{1, 2} = \frac{- ( - 73.5 ) \pm ( - 73.5 ) ^{2} - 4 ( - 3355.2025 ) \cdot 2453.701875}{2 ( - 3355.2025 )}

u_{1, 2} = \frac{- ( - 73.5 ) \pm ( - 73.5 ) ^{2} - 4 ( - 3355.2025 ) \cdot 2453.701875}{2 ( - 3355.2025 )}

(- 73.5)^{2} - 4 (- 3355.2025) \cdot 2453.701875 = 32936068.91101 \dots

(- 73.5)^{2} - 4 (- 3355.2025) \cdot 2453.701875

Apply rule

- (- a) = a

= (- 73.5)^{2} + 4 \cdot 3355.2025 \cdot 2453.701875

Apply exponent rule:

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

n

is even

(- 73.5)^{2} = 73. 5^{2}

= 73. 5^{2} + 4 \cdot 2453.701875 \cdot 3355.2025

Multiply the numbers:

4 \cdot 3355.2025 \cdot 2453.701875 = 32930666.66101 \dots

= 73. 5^{2} + 32930666.66101 \dots

73. 5^{2} = 5402.25

= 5402.25 + 32930666.66101 \dots

Add the numbers:

5402.25 + 32930666.66101 \dots = 32936068.91101 \dots

= 32936068.91101 \dots

u_{1, 2} = \frac{- ( - 73.5 ) \pm 32936068.91101 \dots}{2 ( - 3355.2025 )}

Separate the solutions

u_{1} = \frac{- ( - 73.5 ) + 32936068.91101 \dots}{2 ( - 3355.2025 )}, u_{2} = \frac{- ( - 73.5 ) - 32936068.91101 \dots}{2 ( - 3355.2025 )}

u = \frac{- ( - 73.5 ) + 32936068.91101 \dots}{2 ( - 3355.2025 )} : - \frac{73.5 + 32936068.91101 \dots}{6710.405}

\frac{- ( - 73.5 ) + 32936068.91101 \dots}{2 ( - 3355.2025 )}

Remove parentheses:

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

= \frac{73.5 + 32936068.91101 \dots}{- 2 \cdot 3355.2025}

Multiply the numbers:

2 \cdot 3355.2025 = 6710.405

= \frac{73.5 + 32936068.91101 \dots}{- 6710.405}

Apply the fraction rule:

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

= - \frac{73.5 + 32936068.91101 \dots}{6710.405}

u = \frac{- ( - 73.5 ) - 32936068.91101 \dots}{2 ( - 3355.2025 )} : \frac{32936068.91101 \dots - 73.5}{6710.405}

\frac{- ( - 73.5 ) - 32936068.91101 \dots}{2 ( - 3355.2025 )}

Remove parentheses:

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

= \frac{73.5 - 32936068.91101 \dots}{- 2 \cdot 3355.2025}

Multiply the numbers:

2 \cdot 3355.2025 = 6710.405

= \frac{73.5 - 32936068.91101 \dots}{- 6710.405}

Apply the fraction rule:

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

73.5 - 32936068.91101 \dots = - (32936068.91101 \dots - 73.5)

= \frac{32936068.91101 \dots - 73.5}{6710.405}

The solutions to the quadratic equation are:

u = - \frac{73.5 + 32936068.91101 \dots}{6710.405}, u = \frac{32936068.91101 \dots - 73.5}{6710.405}

Substitute back

u = sin (θ)

sin (θ) = - \frac{73.5 + 32936068.91101 \dots}{6710.405}, sin (θ) = \frac{32936068.91101 \dots - 73.5}{6710.405}

sin (θ) = - \frac{73.5 + 32936068.91101 \dots}{6710.405}, sin (θ) = \frac{32936068.91101 \dots - 73.5}{6710.405}

sin (θ) = - \frac{73.5 + 32936068.91101 \dots}{6710.405} : θ = arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn, θ = π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn

sin (θ) = - \frac{73.5 + 32936068.91101 \dots}{6710.405}

Apply trig inverse properties

sin (θ) = - \frac{73.5 + 32936068.91101 \dots}{6710.405}

General solutions for

sin (θ) = - \frac{73.5 + 32936068.91101 \dots}{6710.405}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

θ = arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn, θ = π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn

θ = arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn, θ = π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn

sin (θ) = \frac{32936068.91101 \dots - 73.5}{6710.405} : θ = arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn, θ = π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn

sin (θ) = \frac{32936068.91101 \dots - 73.5}{6710.405}

Apply trig inverse properties

sin (θ) = \frac{32936068.91101 \dots - 73.5}{6710.405}

General solutions for

sin (θ) = \frac{32936068.91101 \dots - 73.5}{6710.405}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

θ = arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn, θ = π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn

θ = arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn, θ = π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn

Combine all the solutions

θ = arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn, θ = π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn, θ = arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn, θ = π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

49.55 cos (θ) - 30 sin (θ) = 1.225

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

Check the solution

arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn :

False

arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn

Plug in

n = 1

arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 π 1

For

49.55 cos (θ) - 30 sin (θ) = 1.225

plug in

θ = arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 π 1

49.55 cos (arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 π 1) - 30 sin (arcsin (- \frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 π 1) = 1.225

Refine

50.74648 \dots = 1.225

\Rightarrow False

Check the solution

π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn :

True

π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 π 1

For

49.55 cos (θ) - 30 sin (θ) = 1.225

plug in

θ = π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 π 1

49.55 cos (π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 π 1) - 30 sin (π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 π 1) = 1.225

Refine

1.22499 \dots = 1.225

\Rightarrow True

Check the solution

arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn :

True

arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn

Plug in

n = 1

arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 π 1

For

49.55 cos (θ) - 30 sin (θ) = 1.225

plug in

θ = arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 π 1

49.55 cos (arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 π 1) - 30 sin (arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 π 1) = 1.225

Refine

1.225 = 1.225

\Rightarrow True

Check the solution

π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn :

False

π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 π 1

For

49.55 cos (θ) - 30 sin (θ) = 1.225

plug in

θ = π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 π 1

49.55 cos (π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 π 1) - 30 sin (π - arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 π 1) = 1.225

Refine

- 51.88211 \dots = 1.225

\Rightarrow False

θ = π + arcsin (\frac{73.5 + 32936068.91101 \dots}{6710.405}) + 2 πn, θ = arcsin (\frac{32936068.91101 \dots - 73.5}{6710.405}) + 2 πn

Show solutions in decimal form

θ = π + 1.04752 \dots + 2 πn, θ = 1.00522 \dots + 2 πn

Graph

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for 49.55cos(θ)-30sin(θ)=1.225 ?
The general solution for 49.55cos(θ)-30sin(θ)=1.225 is θ=pi+1.04752…+2pin,θ=1.00522…+2pin