What is the general solution for 4-7sin(x)=cos^2(x) ?

The general solution for 4-7sin(x)=cos^2(x) is x=0.47644…+2pin,x=pi-0.47644…+2pin

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Popular Trigonometry >

4-7sin(x)=cos^2(x)

Solution

4 - 7 sin (x) = cos^{2} (x)

Solution

x = 0.47644 \dots + 2 πn, x = π - 0.47644 \dots + 2 πn

Degrees

x = 27.29801 \dots^{\circ} + 36 0^{\circ} n, x = 152.70198 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 - 7 sin (x) = cos^{2} (x)

Subtract

cos^{2} (x)

from both sides

4 - 7 sin (x) - cos^{2} (x) = 0

Rewrite using trig identities

4 - cos^{2} (x) - 7 sin (x)

Use the Pythagorean identity:

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

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

= 4 - (1 - sin^{2} (x)) - 7 sin (x)

Simplify

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

= 4 - 1 + sin^{2} (x) - 7 sin (x)

Subtract the numbers:

4 - 1 = 3

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

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

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

Solve by substitution

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

Let:

sin (x) = u

3 + u^{2} - 7 u = 0

3 + u^{2} - 7 u = 0 : u = \frac{7 + 37}{2}, u = \frac{7 - 37}{2}

3 + u^{2} - 7 u = 0

Write in the standard form

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

u^{2} - 7 u + 3 = 0

Solve with the quadratic formula

u^{2} - 7 u + 3 = 0

Quadratic Equation Formula:

For

a = 1, b = - 7, c = 3

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

(- 7)^{2} - 4 \cdot 1 \cdot 3 = 37

(- 7)^{2} - 4 \cdot 1 \cdot 3

Apply exponent rule:

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

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 1 \cdot 3

Multiply the numbers:

4 \cdot 1 \cdot 3 = 12

= 7^{2} - 12

7^{2} = 49

= 49 - 12

Subtract the numbers:

49 - 12 = 37

= 37

u_{1, 2} = \frac{- ( - 7 ) \pm 37}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 37}{2 \cdot 1}, u_{2} = \frac{- ( - 7 ) - 37}{2 \cdot 1}

u = \frac{- ( - 7 ) + 37}{2 \cdot 1} : \frac{7 + 37}{2}

\frac{- ( - 7 ) + 37}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{7 + 37}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{7 + 37}{2}

u = \frac{- ( - 7 ) - 37}{2 \cdot 1} : \frac{7 - 37}{2}

\frac{- ( - 7 ) - 37}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{7 - 37}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{7 - 37}{2}

The solutions to the quadratic equation are:

u = \frac{7 + 37}{2}, u = \frac{7 - 37}{2}

Substitute back

u = sin (x)

sin (x) = \frac{7 + 37}{2}, sin (x) = \frac{7 - 37}{2}

sin (x) = \frac{7 + 37}{2}, sin (x) = \frac{7 - 37}{2}

sin (x) = \frac{7 + 37}{2} :

No Solution

sin (x) = \frac{7 + 37}{2}

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = \frac{7 - 37}{2} : x = arcsin (\frac{7 - 37}{2}) + 2 πn, x = π - arcsin (\frac{7 - 37}{2}) + 2 πn

sin (x) = \frac{7 - 37}{2}

Apply trig inverse properties

sin (x) = \frac{7 - 37}{2}

General solutions for

sin (x) = \frac{7 - 37}{2}

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

x = arcsin (\frac{7 - 37}{2}) + 2 πn, x = π - arcsin (\frac{7 - 37}{2}) + 2 πn

x = arcsin (\frac{7 - 37}{2}) + 2 πn, x = π - arcsin (\frac{7 - 37}{2}) + 2 πn

Combine all the solutions

x = arcsin (\frac{7 - 37}{2}) + 2 πn, x = π - arcsin (\frac{7 - 37}{2}) + 2 πn

Show solutions in decimal form

x = 0.47644 \dots + 2 πn, x = π - 0.47644 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4-7sin(x)=cos^2(x) ?
The general solution for 4-7sin(x)=cos^2(x) is x=0.47644…+2pin,x=pi-0.47644…+2pin