What is the general solution for 2-3sin(θ)=cos(θ) ?

The general solution for 2-3sin(θ)=cos(θ) is θ=pi-1.00646…+2pin,θ=0.36296…+2pin

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Popular Trigonometry >

2-3sin(θ)=cos(θ)

Solution

2 - 3 sin (θ) = cos (θ)

Solution

θ = π - 1.00646 \dots + 2 πn, θ = 0.36296 \dots + 2 πn

Degrees

θ = 122.33353 \dots^{\circ} + 36 0^{\circ} n, θ = 20.79657 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 - 3 sin (θ) = cos (θ)

Square both sides

(2 - 3 sin (θ))^{2} = cos^{2} (θ)

Subtract

cos^{2} (θ)

from both sides

(2 - 3 sin (θ))^{2} - cos^{2} (θ) = 0

Rewrite using trig identities

(2 - 3 sin (θ))^{2} - cos^{2} (θ)

Use the Pythagorean identity:

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

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

= (2 - 3 sin (θ))^{2} - (1 - sin^{2} (θ))

Simplify

(2 - 3 sin (θ))^{2} - (1 - sin^{2} (θ)) : 10 sin^{2} (θ) - 12 sin (θ) + 3

(2 - 3 sin (θ))^{2} - (1 - sin^{2} (θ))

(2 - 3 sin (θ))^{2} : 4 - 12 sin (θ) + 9 sin^{2} (θ)

Apply Perfect Square Formula:

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

a = 2, b = 3 sin (θ)

= 2^{2} - 2 \cdot 2 \cdot 3 sin (θ) + (3 sin (θ))^{2}

Simplify

2^{2} - 2 \cdot 2 \cdot 3 sin (θ) + (3 sin (θ))^{2} : 4 - 12 sin (θ) + 9 sin^{2} (θ)

2^{2} - 2 \cdot 2 \cdot 3 sin (θ) + (3 sin (θ))^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 \cdot 3 sin (θ) = 12 sin (θ)

2 \cdot 2 \cdot 3 sin (θ)

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12 sin (θ)

(3 sin (θ))^{2} = 9 sin^{2} (θ)

(3 sin (θ))^{2}

Apply exponent rule:

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

= 3^{2} sin^{2} (θ)

3^{2} = 9

= 9 sin^{2} (θ)

= 4 - 12 sin (θ) + 9 sin^{2} (θ)

= 4 - 12 sin (θ) + 9 sin^{2} (θ)

= 4 - 12 sin (θ) + 9 sin^{2} (θ) - (1 - sin^{2} (θ))

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

- (1 - sin^{2} (θ))

Distribute parentheses

= - (1) - (- sin^{2} (θ))

Apply minus-plus rules

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

= - 1 + sin^{2} (θ)

= 4 - 12 sin (θ) + 9 sin^{2} (θ) - 1 + sin^{2} (θ)

Simplify

4 - 12 sin (θ) + 9 sin^{2} (θ) - 1 + sin^{2} (θ) : 10 sin^{2} (θ) - 12 sin (θ) + 3

4 - 12 sin (θ) + 9 sin^{2} (θ) - 1 + sin^{2} (θ)

Group like terms

= - 12 sin (θ) + 9 sin^{2} (θ) + sin^{2} (θ) + 4 - 1

Add similar elements:

9 sin^{2} (θ) + sin^{2} (θ) = 10 sin^{2} (θ)

= - 12 sin (θ) + 10 sin^{2} (θ) + 4 - 1

Add/Subtract the numbers:

4 - 1 = 3

= 10 sin^{2} (θ) - 12 sin (θ) + 3

= 10 sin^{2} (θ) - 12 sin (θ) + 3

= 10 sin^{2} (θ) - 12 sin (θ) + 3

3 + 10 sin^{2} (θ) - 12 sin (θ) = 0

Solve by substitution

3 + 10 sin^{2} (θ) - 12 sin (θ) = 0

Let:

sin (θ) = u

3 + 10 u^{2} - 12 u = 0

3 + 10 u^{2} - 12 u = 0 : u = \frac{6 + 6}{10}, u = \frac{6 - 6}{10}

3 + 10 u^{2} - 12 u = 0

Write in the standard form

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

10 u^{2} - 12 u + 3 = 0

Solve with the quadratic formula

10 u^{2} - 12 u + 3 = 0

Quadratic Equation Formula:

For

a = 10, b = - 12, c = 3

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 10 \cdot 3}{2 \cdot 10}

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 10 \cdot 3}{2 \cdot 10}

(- 12)^{2} - 4 \cdot 10 \cdot 3 = 26

(- 12)^{2} - 4 \cdot 10 \cdot 3

Apply exponent rule:

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

n

is even

(- 12)^{2} = 1 2^{2}

= 1 2^{2} - 4 \cdot 10 \cdot 3

Multiply the numbers:

4 \cdot 10 \cdot 3 = 120

= 1 2^{2} - 120

1 2^{2} = 144

= 144 - 120

Subtract the numbers:

144 - 120 = 24

= 24

Prime factorization of

24 : 2^{3} \cdot 3

24

24

divides by

224 = 12 \cdot 2

= 2 \cdot 12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3

= 2^{3} \cdot 3

= 2^{3} \cdot 3

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2 \cdot 3

Apply radical rule:

= 2^{2} 2 \cdot 3

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 3

Refine

= 26

u_{1, 2} = \frac{- ( - 12 ) \pm 2 6}{2 \cdot 10}

Separate the solutions

u_{1} = \frac{- ( - 12 ) + 2 6}{2 \cdot 10}, u_{2} = \frac{- ( - 12 ) - 2 6}{2 \cdot 10}

u = \frac{- ( - 12 ) + 2 6}{2 \cdot 10} : \frac{6 + 6}{10}

\frac{- ( - 12 ) + 2 6}{2 \cdot 10}

Apply rule

- (- a) = a

= \frac{12 + 2 6}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{12 + 2 6}{20}

Factor

12 + 26 : 2 (6 + 6)

12 + 26

Rewrite as

= 2 \cdot 6 + 26

Factor out common term

2

= 2 (6 + 6)

= \frac{2 ( 6 + 6 )}{20}

Cancel the common factor:

2

= \frac{6 + 6}{10}

u = \frac{- ( - 12 ) - 2 6}{2 \cdot 10} : \frac{6 - 6}{10}

\frac{- ( - 12 ) - 2 6}{2 \cdot 10}

Apply rule

- (- a) = a

= \frac{12 - 2 6}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{12 - 2 6}{20}

Factor

12 - 26 : 2 (6 - 6)

12 - 26

Rewrite as

= 2 \cdot 6 - 26

Factor out common term

2

= 2 (6 - 6)

= \frac{2 ( 6 - 6 )}{20}

Cancel the common factor:

2

= \frac{6 - 6}{10}

The solutions to the quadratic equation are:

u = \frac{6 + 6}{10}, u = \frac{6 - 6}{10}

Substitute back

u = sin (θ)

sin (θ) = \frac{6 + 6}{10}, sin (θ) = \frac{6 - 6}{10}

sin (θ) = \frac{6 + 6}{10}, sin (θ) = \frac{6 - 6}{10}

sin (θ) = \frac{6 + 6}{10} : θ = arcsin (\frac{6 + 6}{10}) + 2 πn, θ = π - arcsin (\frac{6 + 6}{10}) + 2 πn

sin (θ) = \frac{6 + 6}{10}

Apply trig inverse properties

sin (θ) = \frac{6 + 6}{10}

General solutions for

sin (θ) = \frac{6 + 6}{10}

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

θ = arcsin (\frac{6 + 6}{10}) + 2 πn, θ = π - arcsin (\frac{6 + 6}{10}) + 2 πn

θ = arcsin (\frac{6 + 6}{10}) + 2 πn, θ = π - arcsin (\frac{6 + 6}{10}) + 2 πn

sin (θ) = \frac{6 - 6}{10} : θ = arcsin (\frac{6 - 6}{10}) + 2 πn, θ = π - arcsin (\frac{6 - 6}{10}) + 2 πn

sin (θ) = \frac{6 - 6}{10}

Apply trig inverse properties

sin (θ) = \frac{6 - 6}{10}

General solutions for

sin (θ) = \frac{6 - 6}{10}

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

θ = arcsin (\frac{6 - 6}{10}) + 2 πn, θ = π - arcsin (\frac{6 - 6}{10}) + 2 πn

θ = arcsin (\frac{6 - 6}{10}) + 2 πn, θ = π - arcsin (\frac{6 - 6}{10}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{6 + 6}{10}) + 2 πn, θ = π - arcsin (\frac{6 + 6}{10}) + 2 πn, θ = arcsin (\frac{6 - 6}{10}) + 2 πn, θ = π - arcsin (\frac{6 - 6}{10}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

2 - 3 sin (θ) = cos (θ)

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

Check the solution

arcsin (\frac{6 + 6}{10}) + 2 πn :

False

arcsin (\frac{6 + 6}{10}) + 2 πn

Plug in

n = 1

arcsin (\frac{6 + 6}{10}) + 2 π 1

For

2 - 3 sin (θ) = cos (θ)

plug in

θ = arcsin (\frac{6 + 6}{10}) + 2 π 1

2 - 3 sin (arcsin (\frac{6 + 6}{10}) + 2 π 1) = cos (arcsin (\frac{6 + 6}{10}) + 2 π 1)

Refine

- 0.53484 \dots = 0.53484 \dots

\Rightarrow False

Check the solution

π - arcsin (\frac{6 + 6}{10}) + 2 πn :

True

π - arcsin (\frac{6 + 6}{10}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{6 + 6}{10}) + 2 π 1

For

2 - 3 sin (θ) = cos (θ)

plug in

θ = π - arcsin (\frac{6 + 6}{10}) + 2 π 1

2 - 3 sin (π - arcsin (\frac{6 + 6}{10}) + 2 π 1) = cos (π - arcsin (\frac{6 + 6}{10}) + 2 π 1)

Refine

- 0.53484 \dots = - 0.53484 \dots

\Rightarrow True

Check the solution

arcsin (\frac{6 - 6}{10}) + 2 πn :

True

arcsin (\frac{6 - 6}{10}) + 2 πn

Plug in

n = 1

arcsin (\frac{6 - 6}{10}) + 2 π 1

For

2 - 3 sin (θ) = cos (θ)

plug in

θ = arcsin (\frac{6 - 6}{10}) + 2 π 1

2 - 3 sin (arcsin (\frac{6 - 6}{10}) + 2 π 1) = cos (arcsin (\frac{6 - 6}{10}) + 2 π 1)

Refine

0.93484 \dots = 0.93484 \dots

\Rightarrow True

Check the solution

π - arcsin (\frac{6 - 6}{10}) + 2 πn :

False

π - arcsin (\frac{6 - 6}{10}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{6 - 6}{10}) + 2 π 1

For

2 - 3 sin (θ) = cos (θ)

plug in

θ = π - arcsin (\frac{6 - 6}{10}) + 2 π 1

2 - 3 sin (π - arcsin (\frac{6 - 6}{10}) + 2 π 1) = cos (π - arcsin (\frac{6 - 6}{10}) + 2 π 1)

Refine

0.93484 \dots = - 0.93484 \dots

\Rightarrow False

θ = π - arcsin (\frac{6 + 6}{10}) + 2 πn, θ = arcsin (\frac{6 - 6}{10}) + 2 πn

Show solutions in decimal form

θ = π - 1.00646 \dots + 2 πn, θ = 0.36296 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2-3sin(θ)=cos(θ) ?
The general solution for 2-3sin(θ)=cos(θ) is θ=pi-1.00646…+2pin,θ=0.36296…+2pin