What is the general solution for 3cos^2(x)=7-8sin(x) ?

The general solution for 3cos^2(x)=7-8sin(x) is x=0.72972…+2pin,x=pi-0.72972…+2pin

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Popular >

3cos^2(x)=7-8sin(x)

Solution

3 cos^{2} (x) = 7 - 8 sin (x)

Solution

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Degrees

x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 cos^{2} (x) = 7 - 8 sin (x)

Subtract

7 - 8 sin (x)

from both sides

3 cos^{2} (x) - 7 + 8 sin (x) = 0

Rewrite using trig identities

- 7 + 3 cos^{2} (x) + 8 sin (x)

Use the Pythagorean identity:

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

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

= - 7 + 3 (1 - sin^{2} (x)) + 8 sin (x)

Simplify

- 7 + 3 (1 - sin^{2} (x)) + 8 sin (x) : 8 sin (x) - 3 sin^{2} (x) - 4

- 7 + 3 (1 - sin^{2} (x)) + 8 sin (x)

Expand

3 (1 - sin^{2} (x)) : 3 - 3 sin^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

= 3 \cdot 1 - 3 sin^{2} (x)

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 sin^{2} (x)

= - 7 + 3 - 3 sin^{2} (x) + 8 sin (x)

Add/Subtract the numbers:

- 7 + 3 = - 4

= 8 sin (x) - 3 sin^{2} (x) - 4

= 8 sin (x) - 3 sin^{2} (x) - 4

- 4 - 3 sin^{2} (x) + 8 sin (x) = 0

Solve by substitution

- 4 - 3 sin^{2} (x) + 8 sin (x) = 0

Let:

sin (x) = u

- 4 - 3 u^{2} + 8 u = 0

- 4 - 3 u^{2} + 8 u = 0 : u = \frac{2}{3}, u = 2

- 4 - 3 u^{2} + 8 u = 0

Write in the standard form

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

- 3 u^{2} + 8 u - 4 = 0

Solve with the quadratic formula

- 3 u^{2} + 8 u - 4 = 0

Quadratic Equation Formula:

For

a = - 3, b = 8, c = - 4

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) ( - 4 )}{2 ( - 3 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) ( - 4 )}{2 ( - 3 )}

8^{2} - 4 (- 3) (- 4) = 4

8^{2} - 4 (- 3) (- 4)

Apply rule

- (- a) = a

= 8^{2} - 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

Subtract the numbers:

64 - 48 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

u_{1, 2} = \frac{- 8 \pm 4}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- 8 + 4}{2 ( - 3 )}, u_{2} = \frac{- 8 - 4}{2 ( - 3 )}

u = \frac{- 8 + 4}{2 ( - 3 )} : \frac{2}{3}

\frac{- 8 + 4}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 4}{- 2 \cdot 3}

Add/Subtract the numbers:

- 8 + 4 = - 4

= \frac{- 4}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4}{- 6}

Apply the fraction rule:

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

= \frac{4}{6}

Cancel the common factor:

2

= \frac{2}{3}

u = \frac{- 8 - 4}{2 ( - 3 )} : 2

\frac{- 8 - 4}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 4}{- 2 \cdot 3}

Subtract the numbers:

- 8 - 4 = - 12

= \frac{- 12}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{- 6}

Apply the fraction rule:

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

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

The solutions to the quadratic equation are:

u = \frac{2}{3}, u = 2

Substitute back

u = sin (x)

sin (x) = \frac{2}{3}, sin (x) = 2

sin (x) = \frac{2}{3}, sin (x) = 2

sin (x) = \frac{2}{3} : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

sin (x) = \frac{2}{3}

Apply trig inverse properties

sin (x) = \frac{2}{3}

General solutions for

sin (x) = \frac{2}{3}

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

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

sin (x) = 2 :

No Solution

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Popular Examples

sin(X)=-1/2

sin (X) = - \frac{1}{2}

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5cos(A)+8=3cos(A)+6

5 cos (A) + 8 = 3 cos (A) + 6

2+7cos(θ)=4sin^2(θ),-180<= x<= 180

2 + 7 cos (θ) = 4 sin^{2} (θ), - 18 0^{\circ} \leq x \leq 18 0^{\circ}

1=2cos(3θ)

1 = 2 cos (3 θ)

Frequently Asked Questions (FAQ)

What is the general solution for 3cos^2(x)=7-8sin(x) ?
The general solution for 3cos^2(x)=7-8sin(x) is x=0.72972…+2pin,x=pi-0.72972…+2pin