What is the general solution for 55sin(θ)-20-20cos(θ)=0 ?

The general solution for 55sin(θ)-20-20cos(θ)=0 is θ=0.69754…+2pin,θ=pi+2pin

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55sin(θ)-20-20cos(θ)=0

Solution

55 sin (θ) - 20 - 20 cos (θ) = 0

Solution

θ = 0.69754 \dots + 2 πn, θ = π + 2 πn

Degrees

θ = 39.96621 \dots^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

55 sin (θ) - 20 - 20 cos (θ) = 0

Add

20 cos (θ)

to both sides

55 sin (θ) - 20 = 20 cos (θ)

Square both sides

(55 sin (θ) - 20)^{2} = (20 cos (θ))^{2}

Subtract

(20 cos (θ))^{2}

from both sides

(55 sin (θ) - 20)^{2} - 400 cos^{2} (θ) = 0

Rewrite using trig identities

(- 20 + 55 sin (θ))^{2} - 400 cos^{2} (θ)

Use the Pythagorean identity:

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

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

= (- 20 + 55 sin (θ))^{2} - 400 (1 - sin^{2} (θ))

Simplify

(- 20 + 55 sin (θ))^{2} - 400 (1 - sin^{2} (θ)) : 3425 sin^{2} (θ) - 2200 sin (θ)

(- 20 + 55 sin (θ))^{2} - 400 (1 - sin^{2} (θ))

(- 20 + 55 sin (θ))^{2} : 400 - 2200 sin (θ) + 3025 sin^{2} (θ)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = - 20, b = 55 sin (θ)

= (- 20)^{2} + 2 (- 20) \cdot 55 sin (θ) + (55 sin (θ))^{2}

Simplify

(- 20)^{2} + 2 (- 20) \cdot 55 sin (θ) + (55 sin (θ))^{2} : 400 - 2200 sin (θ) + 3025 sin^{2} (θ)

(- 20)^{2} + 2 (- 20) \cdot 55 sin (θ) + (55 sin (θ))^{2}

Remove parentheses:

(- a) = - a

= (- 20)^{2} - 2 \cdot 20 \cdot 55 sin (θ) + (55 sin (θ))^{2}

(- 20)^{2} = 400

(- 20)^{2}

Apply exponent rule:

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

n

is even

(- 20)^{2} = 2 0^{2}

= 2 0^{2}

2 0^{2} = 400

= 400

2 \cdot 20 \cdot 55 sin (θ) = 2200 sin (θ)

2 \cdot 20 \cdot 55 sin (θ)

Multiply the numbers:

2 \cdot 20 \cdot 55 = 2200

= 2200 sin (θ)

(55 sin (θ))^{2} = 3025 sin^{2} (θ)

(55 sin (θ))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 5 5^{2} sin^{2} (θ)

5 5^{2} = 3025

= 3025 sin^{2} (θ)

= 400 - 2200 sin (θ) + 3025 sin^{2} (θ)

= 400 - 2200 sin (θ) + 3025 sin^{2} (θ)

= 400 - 2200 sin (θ) + 3025 sin^{2} (θ) - 400 (1 - sin^{2} (θ))

Expand

- 400 (1 - sin^{2} (θ)) : - 400 + 400 sin^{2} (θ)

- 400 (1 - sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = - 400, b = 1, c = sin^{2} (θ)

= - 400 \cdot 1 - (- 400) sin^{2} (θ)

Apply minus-plus rules

- (- a) = a

= - 400 \cdot 1 + 400 sin^{2} (θ)

Multiply the numbers:

400 \cdot 1 = 400

= - 400 + 400 sin^{2} (θ)

= 400 - 2200 sin (θ) + 3025 sin^{2} (θ) - 400 + 400 sin^{2} (θ)

Simplify

400 - 2200 sin (θ) + 3025 sin^{2} (θ) - 400 + 400 sin^{2} (θ) : 3425 sin^{2} (θ) - 2200 sin (θ)

400 - 2200 sin (θ) + 3025 sin^{2} (θ) - 400 + 400 sin^{2} (θ)

Group like terms

= - 2200 sin (θ) + 3025 sin^{2} (θ) + 400 sin^{2} (θ) + 400 - 400

Add similar elements:

3025 sin^{2} (θ) + 400 sin^{2} (θ) = 3425 sin^{2} (θ)

= - 2200 sin (θ) + 3425 sin^{2} (θ) + 400 - 400

400 - 400 = 0

= 3425 sin^{2} (θ) - 2200 sin (θ)

= 3425 sin^{2} (θ) - 2200 sin (θ)

= 3425 sin^{2} (θ) - 2200 sin (θ)

- 2200 sin (θ) + 3425 sin^{2} (θ) = 0

Solve by substitution

- 2200 sin (θ) + 3425 sin^{2} (θ) = 0

Let:

sin (θ) = u

- 2200 u + 3425 u^{2} = 0

- 2200 u + 3425 u^{2} = 0 : u = \frac{88}{137}, u = 0

- 2200 u + 3425 u^{2} = 0

Divide both sides by

3425

- \frac{2200 u}{3425} + \frac{3425 u ^{2}}{3425} = \frac{0}{3425}

Write in the standard form

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

u^{2} - \frac{88 u}{137} = 0

Solve with the quadratic formula

u^{2} - \frac{88 u}{137} = 0

Quadratic Equation Formula:

For

a = 1, b = - \frac{88}{137}, c = 0

u_{1, 2} = \frac{- ( - \frac{88}{137} ) \pm ( - \frac{88}{137} ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - \frac{88}{137} ) \pm ( - \frac{88}{137} ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- \frac{88}{137})^{2} - 4 \cdot 1 \cdot 0 = \frac{88}{137}

(- \frac{88}{137})^{2} - 4 \cdot 1 \cdot 0

(- \frac{88}{137})^{2} = \frac{8 8 ^{2}}{13 7 ^{2}}

(- \frac{88}{137})^{2}

Apply exponent rule:

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

n

is even

(- \frac{88}{137})^{2} = (\frac{88}{137})^{2}

= (\frac{88}{137})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{8 8 ^{2}}{13 7 ^{2}}

4 \cdot 1 \cdot 0 = 0

4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= \frac{8 8 ^{2}}{13 7 ^{2}} - 0

\frac{8 8 ^{2}}{13 7 ^{2}} - 0 = \frac{8 8 ^{2}}{13 7 ^{2}}

= \frac{8 8 ^{2}}{13 7 ^{2}}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{8 8 ^{2}}{13 7 ^{2}}

Apply radical rule:

assuming

a \geq 0

13 7^{2} = 137

= \frac{8 8 ^{2}}{137}

Apply radical rule:

assuming

a \geq 0

8 8^{2} = 88

= \frac{88}{137}

u_{1, 2} = \frac{- ( - \frac{88}{137} ) \pm \frac{88}{137}}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - \frac{88}{137} ) + \frac{88}{137}}{2 \cdot 1}, u_{2} = \frac{- ( - \frac{88}{137} ) - \frac{88}{137}}{2 \cdot 1}

u = \frac{- ( - \frac{88}{137} ) + \frac{88}{137}}{2 \cdot 1} : \frac{88}{137}

\frac{- ( - \frac{88}{137} ) + \frac{88}{137}}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{\frac{88}{137} + \frac{88}{137}}{2 \cdot 1}

Add similar elements:

\frac{88}{137} + \frac{88}{137} = 2 \cdot \frac{88}{137}

= \frac{2 \cdot \frac{88}{137}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 \cdot \frac{88}{137}}{2}

Multiply

2 \cdot \frac{88}{137} : \frac{176}{137}

2 \cdot \frac{88}{137}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{88 \cdot 2}{137}

Multiply the numbers:

88 \cdot 2 = 176

= \frac{176}{137}

= \frac{\frac{176}{137}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{176}{137 \cdot 2}

Multiply the numbers:

137 \cdot 2 = 274

= \frac{176}{274}

Cancel the common factor:

2

= \frac{88}{137}

u = \frac{- ( - \frac{88}{137} ) - \frac{88}{137}}{2 \cdot 1} : 0

\frac{- ( - \frac{88}{137} ) - \frac{88}{137}}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{\frac{88}{137} - \frac{88}{137}}{2 \cdot 1}

Add similar elements:

\frac{88}{137} - \frac{88}{137} = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = \frac{88}{137}, u = 0

Substitute back

u = sin (θ)

sin (θ) = \frac{88}{137}, sin (θ) = 0

sin (θ) = \frac{88}{137}, sin (θ) = 0

sin (θ) = \frac{88}{137} : θ = arcsin (\frac{88}{137}) + 2 πn, θ = π - arcsin (\frac{88}{137}) + 2 πn

sin (θ) = \frac{88}{137}

Apply trig inverse properties

sin (θ) = \frac{88}{137}

General solutions for

sin (θ) = \frac{88}{137}

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

θ = arcsin (\frac{88}{137}) + 2 πn, θ = π - arcsin (\frac{88}{137}) + 2 πn

θ = arcsin (\frac{88}{137}) + 2 πn, θ = π - arcsin (\frac{88}{137}) + 2 πn

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

Combine all the solutions

θ = arcsin (\frac{88}{137}) + 2 πn, θ = π - arcsin (\frac{88}{137}) + 2 πn, θ = 2 πn, θ = π + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

55 sin (θ) - 20 - 20 cos (θ) = 0

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

Check the solution

arcsin (\frac{88}{137}) + 2 πn :

True

arcsin (\frac{88}{137}) + 2 πn

Plug in

n = 1

arcsin (\frac{88}{137}) + 2 π 1

For

55 sin (θ) - 20 - 20 cos (θ) = 0

plug in

θ = arcsin (\frac{88}{137}) + 2 π 1

55 sin (arcsin (\frac{88}{137}) + 2 π 1) - 20 - 20 cos (arcsin (\frac{88}{137}) + 2 π 1) = 0

Refine

0 = 0

\Rightarrow True

Check the solution

π - arcsin (\frac{88}{137}) + 2 πn :

False

π - arcsin (\frac{88}{137}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{88}{137}) + 2 π 1

For

55 sin (θ) - 20 - 20 cos (θ) = 0

plug in

θ = π - arcsin (\frac{88}{137}) + 2 π 1

55 sin (π - arcsin (\frac{88}{137}) + 2 π 1) - 20 - 20 cos (π - arcsin (\frac{88}{137}) + 2 π 1) = 0

Refine

30.65693 \dots = 0

\Rightarrow False

Check the solution

2 πn :

False

2 πn

Plug in

n = 1

2 π 1

For

55 sin (θ) - 20 - 20 cos (θ) = 0

plug in

θ = 2 π 1

55 sin (2 π 1) - 20 - 20 cos (2 π 1) = 0

Refine

- 40 = 0

\Rightarrow False

Check the solution

π + 2 πn :

True

π + 2 πn

Plug in

n = 1

π + 2 π 1

For

55 sin (θ) - 20 - 20 cos (θ) = 0

plug in

θ = π + 2 π 1

55 sin (π + 2 π 1) - 20 - 20 cos (π + 2 π 1) = 0

Refine

0 = 0

\Rightarrow True

θ = arcsin (\frac{88}{137}) + 2 πn, θ = π + 2 πn

Show solutions in decimal form

θ = 0.69754 \dots + 2 πn, θ = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 55sin(θ)-20-20cos(θ)=0 ?
The general solution for 55sin(θ)-20-20cos(θ)=0 is θ=0.69754…+2pin,θ=pi+2pin