What is the general solution for 4cos^2(x)+17sin(x)=8 ?

The general solution for 4cos^2(x)+17sin(x)=8 is x=0.25268…+2pin,x=pi-0.25268…+2pin

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4cos^2(x)+17sin(x)=8

Solution

4 cos^{2} (x) + 17 sin (x) = 8

Solution

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

Degrees

x = 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 165.52248 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 cos^{2} (x) + 17 sin (x) = 8

Subtract

8

from both sides

4 cos^{2} (x) + 17 sin (x) - 8 = 0

Rewrite using trig identities

- 8 + 17 sin (x) + 4 cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 8 + 17 sin (x) + 4 (1 - sin^{2} (x))

Simplify

- 8 + 17 sin (x) + 4 (1 - sin^{2} (x)) : 17 sin (x) - 4 sin^{2} (x) - 4

- 8 + 17 sin (x) + 4 (1 - sin^{2} (x))

Expand

4 (1 - sin^{2} (x)) : 4 - 4 sin^{2} (x)

4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

= 4 \cdot 1 - 4 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

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

= - 8 + 17 sin (x) + 4 - 4 sin^{2} (x)

Simplify

- 8 + 17 sin (x) + 4 - 4 sin^{2} (x) : 17 sin (x) - 4 sin^{2} (x) - 4

- 8 + 17 sin (x) + 4 - 4 sin^{2} (x)

Group like terms

= 17 sin (x) - 4 sin^{2} (x) - 8 + 4

Add/Subtract the numbers:

- 8 + 4 = - 4

= 17 sin (x) - 4 sin^{2} (x) - 4

= 17 sin (x) - 4 sin^{2} (x) - 4

= 17 sin (x) - 4 sin^{2} (x) - 4

- 4 + 17 sin (x) - 4 sin^{2} (x) = 0

Solve by substitution

- 4 + 17 sin (x) - 4 sin^{2} (x) = 0

Let:

sin (x) = u

- 4 + 17 u - 4 u^{2} = 0

- 4 + 17 u - 4 u^{2} = 0 : u = \frac{1}{4}, u = 4

- 4 + 17 u - 4 u^{2} = 0

Write in the standard form

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

- 4 u^{2} + 17 u - 4 = 0

Solve with the quadratic formula

- 4 u^{2} + 17 u - 4 = 0

Quadratic Equation Formula:

For

a = - 4, b = 17, c = - 4

u_{1, 2} = \frac{- 17 \pm 1 7 ^{2} - 4 ( - 4 ) ( - 4 )}{2 ( - 4 )}

u_{1, 2} = \frac{- 17 \pm 1 7 ^{2} - 4 ( - 4 ) ( - 4 )}{2 ( - 4 )}

1 7^{2} - 4 (- 4) (- 4) = 15

1 7^{2} - 4 (- 4) (- 4)

Apply rule

- (- a) = a

= 1 7^{2} - 4 \cdot 4 \cdot 4

Multiply the numbers:

4 \cdot 4 \cdot 4 = 64

= 1 7^{2} - 64

1 7^{2} = 289

= 289 - 64

Subtract the numbers:

289 - 64 = 225

= 225

Factor the number:

225 = 1 5^{2}

= 1 5^{2}

Apply radical rule:

1 5^{2} = 15

= 15

u_{1, 2} = \frac{- 17 \pm 15}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 17 + 15}{2 ( - 4 )}, u_{2} = \frac{- 17 - 15}{2 ( - 4 )}

u = \frac{- 17 + 15}{2 ( - 4 )} : \frac{1}{4}

\frac{- 17 + 15}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 17 + 15}{- 2 \cdot 4}

Add/Subtract the numbers:

- 17 + 15 = - 2

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 2}{- 8}

Apply the fraction rule:

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

= \frac{2}{8}

Cancel the common factor:

2

= \frac{1}{4}

u = \frac{- 17 - 15}{2 ( - 4 )} : 4

\frac{- 17 - 15}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 17 - 15}{- 2 \cdot 4}

Subtract the numbers:

- 17 - 15 = - 32

= \frac{- 32}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 32}{- 8}

Apply the fraction rule:

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

= \frac{32}{8}

Divide the numbers:

\frac{32}{8} = 4

= 4

The solutions to the quadratic equation are:

u = \frac{1}{4}, u = 4

Substitute back

u = sin (x)

sin (x) = \frac{1}{4}, sin (x) = 4

sin (x) = \frac{1}{4}, sin (x) = 4

sin (x) = \frac{1}{4} : x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

sin (x) = \frac{1}{4}

Apply trig inverse properties

sin (x) = \frac{1}{4}

General solutions for

sin (x) = \frac{1}{4}

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

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

sin (x) = 4 :

No Solution

sin (x) = 4

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

Show solutions in decimal form

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4cos^2(x)+17sin(x)=8 ?
The general solution for 4cos^2(x)+17sin(x)=8 is x=0.25268…+2pin,x=pi-0.25268…+2pin