What is the general solution for 7sin^2(θ)-36sin(θ)+5=0 ?

The general solution for 7sin^2(θ)-36sin(θ)+5=0 is θ=0.14334…+2pin,θ=pi-0.14334…+2pin

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Popular >

7sin^2(θ)-36sin(θ)+5=0

Solution

7 sin^{2} (θ) - 36 sin (θ) + 5 = 0

Solution

θ = 0.14334 \dots + 2 πn, θ = π - 0.14334 \dots + 2 πn

Degrees

θ = 8.21321 \dots^{\circ} + 36 0^{\circ} n, θ = 171.78678 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 sin^{2} (θ) - 36 sin (θ) + 5 = 0

Solve by substitution

7 sin^{2} (θ) - 36 sin (θ) + 5 = 0

Let:

sin (θ) = u

7 u^{2} - 36 u + 5 = 0

7 u^{2} - 36 u + 5 = 0 : u = 5, u = \frac{1}{7}

7 u^{2} - 36 u + 5 = 0

Solve with the quadratic formula

7 u^{2} - 36 u + 5 = 0

Quadratic Equation Formula:

For

a = 7, b = - 36, c = 5

u_{1, 2} = \frac{- ( - 36 ) \pm ( - 36 ) ^{2} - 4 \cdot 7 \cdot 5}{2 \cdot 7}

u_{1, 2} = \frac{- ( - 36 ) \pm ( - 36 ) ^{2} - 4 \cdot 7 \cdot 5}{2 \cdot 7}

(- 36)^{2} - 4 \cdot 7 \cdot 5 = 34

(- 36)^{2} - 4 \cdot 7 \cdot 5

Apply exponent rule:

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

n

is even

(- 36)^{2} = 3 6^{2}

= 3 6^{2} - 4 \cdot 7 \cdot 5

Multiply the numbers:

4 \cdot 7 \cdot 5 = 140

= 3 6^{2} - 140

3 6^{2} = 1296

= 1296 - 140

Subtract the numbers:

1296 - 140 = 1156

= 1156

Factor the number:

1156 = 3 4^{2}

= 3 4^{2}

Apply radical rule:

3 4^{2} = 34

= 34

u_{1, 2} = \frac{- ( - 36 ) \pm 34}{2 \cdot 7}

Separate the solutions

u_{1} = \frac{- ( - 36 ) + 34}{2 \cdot 7}, u_{2} = \frac{- ( - 36 ) - 34}{2 \cdot 7}

u = \frac{- ( - 36 ) + 34}{2 \cdot 7} : 5

\frac{- ( - 36 ) + 34}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{36 + 34}{2 \cdot 7}

Add the numbers:

36 + 34 = 70

= \frac{70}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{70}{14}

Divide the numbers:

\frac{70}{14} = 5

= 5

u = \frac{- ( - 36 ) - 34}{2 \cdot 7} : \frac{1}{7}

\frac{- ( - 36 ) - 34}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{36 - 34}{2 \cdot 7}

Subtract the numbers:

36 - 34 = 2

= \frac{2}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{2}{14}

Cancel the common factor:

2

= \frac{1}{7}

The solutions to the quadratic equation are:

u = 5, u = \frac{1}{7}

Substitute back

u = sin (θ)

sin (θ) = 5, sin (θ) = \frac{1}{7}

sin (θ) = 5, sin (θ) = \frac{1}{7}

sin (θ) = 5 :

No Solution

sin (θ) = 5

- 1 \leq sin (x) \leq 1

No Solution

sin (θ) = \frac{1}{7} : θ = arcsin (\frac{1}{7}) + 2 πn, θ = π - arcsin (\frac{1}{7}) + 2 πn

sin (θ) = \frac{1}{7}

Apply trig inverse properties

sin (θ) = \frac{1}{7}

General solutions for

sin (θ) = \frac{1}{7}

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

θ = arcsin (\frac{1}{7}) + 2 πn, θ = π - arcsin (\frac{1}{7}) + 2 πn

θ = arcsin (\frac{1}{7}) + 2 πn, θ = π - arcsin (\frac{1}{7}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{1}{7}) + 2 πn, θ = π - arcsin (\frac{1}{7}) + 2 πn

Show solutions in decimal form

θ = 0.14334 \dots + 2 πn, θ = π - 0.14334 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 7sin^2(θ)-36sin(θ)+5=0 ?
The general solution for 7sin^2(θ)-36sin(θ)+5=0 is θ=0.14334…+2pin,θ=pi-0.14334…+2pin