What is the general solution for 10sin(x)+10cos^2(x)=10,0<= x<2pi ?

Q: What is the general solution for 10sin(x)+10cos^2(x)=10,0<= x<2pi ?

The general solution for 10sin(x)+10cos^2(x)=10,0<= x<2pi is x=0,x=pi,x= pi/2

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10sin(x)+10cos^2(x)=10,0<= x<2pi

Solution

10 sin (x) + 10 cos^{2} (x) = 10, 0 \leq x < 2 π

Solution

x = 0, x = π, x = \frac{π}{2}

Degrees

x = 0^{\circ}, x = 18 0^{\circ}, x = 9 0^{\circ}

Solution steps

10 sin (x) + 10 cos^{2} (x) = 10, 0 \leq x < 2 π

Subtract

10

from both sides

10 sin (x) + 10 cos^{2} (x) - 10 = 0

Rewrite using trig identities

- 10 + 10 cos^{2} (x) + 10 sin (x)

Use the Pythagorean identity:

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

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

= - 10 + 10 (1 - sin^{2} (x)) + 10 sin (x)

Simplify

- 10 + 10 (1 - sin^{2} (x)) + 10 sin (x) : 10 sin (x) - 10 sin^{2} (x)

- 10 + 10 (1 - sin^{2} (x)) + 10 sin (x)

Expand

10 (1 - sin^{2} (x)) : 10 - 10 sin^{2} (x)

10 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 10, b = 1, c = sin^{2} (x)

= 10 \cdot 1 - 10 sin^{2} (x)

Multiply the numbers:

10 \cdot 1 = 10

= 10 - 10 sin^{2} (x)

= - 10 + 10 - 10 sin^{2} (x) + 10 sin (x)

- 10 + 10 = 0

= 10 sin (x) - 10 sin^{2} (x)

= 10 sin (x) - 10 sin^{2} (x)

10 sin (x) - 10 sin^{2} (x) = 0

Solve by substitution

10 sin (x) - 10 sin^{2} (x) = 0

Let:

sin (x) = u

10 u - 10 u^{2} = 0

10 u - 10 u^{2} = 0 : u = 0, u = 1

10 u - 10 u^{2} = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 10, b = 10, c = 0

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 ( - 10 ) \cdot 0}{2 ( - 10 )}

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 ( - 10 ) \cdot 0}{2 ( - 10 )}

1 0^{2} - 4 (- 10) \cdot 0 = 10

1 0^{2} - 4 (- 10) \cdot 0

Apply rule

- (- a) = a

= 1 0^{2} + 4 \cdot 10 \cdot 0

Apply rule

0 \cdot a = 0

= 1 0^{2} + 0

1 0^{2} + 0 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

assuming

a \geq 0

= 10

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

Separate the solutions

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

u = \frac{- 10 + 10}{2 ( - 10 )} : 0

\frac{- 10 + 10}{2 ( - 10 )}

Remove parentheses:

(- a) = - a

= \frac{- 10 + 10}{- 2 \cdot 10}

Add/Subtract the numbers:

- 10 + 10 = 0

= \frac{0}{- 2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{0}{- 20}

Apply the fraction rule:

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

= - \frac{0}{20}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

u = \frac{- 10 - 10}{2 ( - 10 )} : 1

\frac{- 10 - 10}{2 ( - 10 )}

Remove parentheses:

(- a) = - a

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

Subtract the numbers:

- 10 - 10 = - 20

= \frac{- 20}{- 2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{- 20}{- 20}

Apply the fraction rule:

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

= \frac{20}{20}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = 0, u = 1

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = 1

sin (x) = 0, sin (x) = 1

sin (x) = 0, 0 \leq x < 2 π : x = 0, x = π

sin (x) = 0, 0 \leq x < 2 π

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Solutions for the range

0 \leq x < 2 π

x = 0, x = π

sin (x) = 1, 0 \leq x < 2 π : x = \frac{π}{2}

sin (x) = 1, 0 \leq x < 2 π

General solutions for

sin (x) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{2}

Combine all the solutions

x = 0, x = π, x = \frac{π}{2}

Popular Examples

sin(x)-sqrt(3-3sin^2(x))=0

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

cos(2x)-cos(x)+1=0

cos (2 x) - cos (x) + 1 = 0

4tan^2(x)+21tan(x)-49=0

4 tan^{2} (x) + 21 tan (x) - 49 = 0

cot(x)=csc(x)+1

cot (x) = csc (x) + 1

(sin(8))/(30)=(sin(x))/(120)

\frac{sin ( 8 )}{30} = \frac{sin ( x )}{120}

Frequently Asked Questions (FAQ)

What is the general solution for 10sin(x)+10cos^2(x)=10,0<= x<2pi ?
The general solution for 10sin(x)+10cos^2(x)=10,0<= x<2pi is x=0,x=pi,x= pi/2