English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

9sin^2(x)+3cos(x)-7=0

Solution

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

Solution

x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Degrees

x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

- 7 + 3 cos (x) + 9 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 7 + 3 cos (x) + 9 (1 - cos^{2} (x))

Simplify

- 7 + 3 cos (x) + 9 (1 - cos^{2} (x)) : 3 cos (x) - 9 cos^{2} (x) + 2

- 7 + 3 cos (x) + 9 (1 - cos^{2} (x))

Expand

9 (1 - cos^{2} (x)) : 9 - 9 cos^{2} (x)

9 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 1, c = cos^{2} (x)

= 9 \cdot 1 - 9 cos^{2} (x)

Multiply the numbers:

9 \cdot 1 = 9

= 9 - 9 cos^{2} (x)

= - 7 + 3 cos (x) + 9 - 9 cos^{2} (x)

Simplify

- 7 + 3 cos (x) + 9 - 9 cos^{2} (x) : 3 cos (x) - 9 cos^{2} (x) + 2

- 7 + 3 cos (x) + 9 - 9 cos^{2} (x)

Group like terms

= 3 cos (x) - 9 cos^{2} (x) - 7 + 9

Add/Subtract the numbers:

- 7 + 9 = 2

= 3 cos (x) - 9 cos^{2} (x) + 2

= 3 cos (x) - 9 cos^{2} (x) + 2

= 3 cos (x) - 9 cos^{2} (x) + 2

2 + 3 cos (x) - 9 cos^{2} (x) = 0

Solve by substitution

2 + 3 cos (x) - 9 cos^{2} (x) = 0

Let:

cos (x) = u

2 + 3 u - 9 u^{2} = 0

2 + 3 u - 9 u^{2} = 0 : u = - \frac{1}{3}, u = \frac{2}{3}

2 + 3 u - 9 u^{2} = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 9, b = 3, c = 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 9 ) \cdot 2}{2 ( - 9 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 9 ) \cdot 2}{2 ( - 9 )}

3^{2} - 4 (- 9) \cdot 2 = 9

3^{2} - 4 (- 9) \cdot 2

Apply rule

- (- a) = a

= 3^{2} + 4 \cdot 9 \cdot 2

Multiply the numbers:

4 \cdot 9 \cdot 2 = 72

= 3^{2} + 72

3^{2} = 9

= 9 + 72

Add the numbers:

9 + 72 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- 3 \pm 9}{2 ( - 9 )}

Separate the solutions

u_{1} = \frac{- 3 + 9}{2 ( - 9 )}, u_{2} = \frac{- 3 - 9}{2 ( - 9 )}

u = \frac{- 3 + 9}{2 ( - 9 )} : - \frac{1}{3}

\frac{- 3 + 9}{2 ( - 9 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 + 9}{- 2 \cdot 9}

Add/Subtract the numbers:

- 3 + 9 = 6

= \frac{6}{- 2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{6}{- 18}

Apply the fraction rule:

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

= - \frac{6}{18}

Cancel the common factor:

6

= - \frac{1}{3}

u = \frac{- 3 - 9}{2 ( - 9 )} : \frac{2}{3}

\frac{- 3 - 9}{2 ( - 9 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 - 9}{- 2 \cdot 9}

Subtract the numbers:

- 3 - 9 = - 12

= \frac{- 12}{- 2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{- 12}{- 18}

Apply the fraction rule:

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

= \frac{12}{18}

Cancel the common factor:

6

= \frac{2}{3}

The solutions to the quadratic equation are:

u = - \frac{1}{3}, u = \frac{2}{3}

Substitute back

u = cos (x)

cos (x) = - \frac{1}{3}, cos (x) = \frac{2}{3}

cos (x) = - \frac{1}{3}, cos (x) = \frac{2}{3}

cos (x) = - \frac{1}{3} : x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

cos (x) = - \frac{1}{3}

Apply trig inverse properties

cos (x) = - \frac{1}{3}

General solutions for

cos (x) = - \frac{1}{3}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

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

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

Combine all the solutions

x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn, x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Graph

Popular Examples

3sec(2x+3)=4

sin(2x)+(sqrt(2))/2 =0

cos(θ)-sin(θ)=sqrt(2)

3sin(2x+30)=tan(2x+30)

16sin(t)cos(t)=4sin(t)