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Popular Trigonometry >

10cos^2(x)=9

Solution

10 cos^{2} (x) = 9

Solution

x = 0.32175 \dots + 2 πn, x = 2 π - 0.32175 \dots + 2 πn, x = 2.81984 \dots + 2 πn, x = - 2.81984 \dots + 2 πn

+1

Degrees

x = 18.43494 \dots^{\circ} + 36 0^{\circ} n, x = 341.56505 \dots^{\circ} + 36 0^{\circ} n, x = 161.56505 \dots^{\circ} + 36 0^{\circ} n, x = - 161.56505 \dots^{\circ} + 36 0^{\circ} n

Solution steps

10 cos^{2} (x) = 9

Solve by substitution

10 cos^{2} (x) = 9

Let:

cos (x) = u

10 u^{2} = 9

10 u^{2} = 9 : u = \frac{3 10}{10}, u = - \frac{3 10}{10}

10 u^{2} = 9

Divide both sides by

10

10 u^{2} = 9

Divide both sides by

10

\frac{10 u ^{2}}{10} = \frac{9}{10}

Simplify

u^{2} = \frac{9}{10}

u^{2} = \frac{9}{10}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{9}{10}, u = - \frac{9}{10}

\frac{9}{10} = \frac{3 10}{10}

\frac{9}{10}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{9}{10}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 3

= \frac{3}{10}

Rationalize

\frac{3}{10} : \frac{3 10}{10}

\frac{3}{10}

Multiply by the conjugate

\frac{10}{10}

= \frac{3 10}{10 10}

1010 = 10

1010

Apply radical rule:

a a = a

1010 = 10

= 10

= \frac{3 10}{10}

= \frac{3 10}{10}

- \frac{9}{10} = - \frac{3 10}{10}

- \frac{9}{10}

Simplify

\frac{9}{10} : \frac{3}{10}

\frac{9}{10}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{9}{10}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 3

= \frac{3}{10}

= - \frac{3}{10}

Rationalize

- \frac{3}{10} : - \frac{3 10}{10}

- \frac{3}{10}

Multiply by the conjugate

\frac{10}{10}

= - \frac{3 10}{10 10}

1010 = 10

1010

Apply radical rule:

a a = a

1010 = 10

= 10

= - \frac{3 10}{10}

= - \frac{3 10}{10}

u = \frac{3 10}{10}, u = - \frac{3 10}{10}

Substitute back

u = cos (x)

cos (x) = \frac{3 10}{10}, cos (x) = - \frac{3 10}{10}

cos (x) = \frac{3 10}{10}, cos (x) = - \frac{3 10}{10}

cos (x) = \frac{3 10}{10} : x = arccos (\frac{3 10}{10}) + 2 πn, x = 2 π - arccos (\frac{3 10}{10}) + 2 πn

cos (x) = \frac{3 10}{10}

Apply trig inverse properties

cos (x) = \frac{3 10}{10}

General solutions for

cos (x) = \frac{3 10}{10}

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

x = arccos (\frac{3 10}{10}) + 2 πn, x = 2 π - arccos (\frac{3 10}{10}) + 2 πn

x = arccos (\frac{3 10}{10}) + 2 πn, x = 2 π - arccos (\frac{3 10}{10}) + 2 πn

cos (x) = - \frac{3 10}{10} : x = arccos (- \frac{3 10}{10}) + 2 πn, x = - arccos (- \frac{3 10}{10}) + 2 πn

cos (x) = - \frac{3 10}{10}

Apply trig inverse properties

cos (x) = - \frac{3 10}{10}

General solutions for

cos (x) = - \frac{3 10}{10}

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

x = arccos (- \frac{3 10}{10}) + 2 πn, x = - arccos (- \frac{3 10}{10}) + 2 πn

x = arccos (- \frac{3 10}{10}) + 2 πn, x = - arccos (- \frac{3 10}{10}) + 2 πn

Combine all the solutions

x = arccos (\frac{3 10}{10}) + 2 πn, x = 2 π - arccos (\frac{3 10}{10}) + 2 πn, x = arccos (- \frac{3 10}{10}) + 2 πn, x = - arccos (- \frac{3 10}{10}) + 2 πn

Show solutions in decimal form

x = 0.32175 \dots + 2 πn, x = 2 π - 0.32175 \dots + 2 πn, x = 2.81984 \dots + 2 πn, x = - 2.81984 \dots + 2 πn

Graph

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