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Popular Trigonometry >

25cos^2(x)=9

Solution

25 cos^{2} (x) = 9

Solution

x = 0.92729 \dots + 2 πn, x = 2 π - 0.92729 \dots + 2 πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

+1

Degrees

x = 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 306.86989 \dots^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = - 126.86989 \dots^{\circ} + 36 0^{\circ} n

Solution steps

25 cos^{2} (x) = 9

Solve by substitution

25 cos^{2} (x) = 9

Let:

cos (x) = u

25 u^{2} = 9

25 u^{2} = 9 : u = \frac{3}{5}, u = - \frac{3}{5}

25 u^{2} = 9

Divide both sides by

25

25 u^{2} = 9

Divide both sides by

25

\frac{25 u ^{2}}{25} = \frac{9}{25}

Simplify

u^{2} = \frac{9}{25}

u^{2} = \frac{9}{25}

For

x^{2} = f (a)

the solutions are

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

u = \frac{9}{25}, u = - \frac{9}{25}

\frac{9}{25} = \frac{3}{5}

\frac{9}{25}

Apply radical rule:

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

assuming

a \geq 0, b \geq 0

= \frac{9}{25}

25 = 5

25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

n a^{n} = a

5^{2} = 5

= 5

= \frac{9}{5}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 3

= \frac{3}{5}

- \frac{9}{25} = - \frac{3}{5}

- \frac{9}{25}

Simplify

\frac{9}{25} : \frac{3}{5}

\frac{9}{25}

Apply radical rule:

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

assuming

a \geq 0, b \geq 0

= \frac{9}{25}

25 = 5

25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

n a^{n} = a

5^{2} = 5

= 5

= \frac{9}{5}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 3

= \frac{3}{5}

= - \frac{3}{5}

u = \frac{3}{5}, u = - \frac{3}{5}

Substitute back

u = cos (x)

cos (x) = \frac{3}{5}, cos (x) = - \frac{3}{5}

cos (x) = \frac{3}{5}, cos (x) = - \frac{3}{5}

cos (x) = \frac{3}{5} : x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn

cos (x) = \frac{3}{5}

Apply trig inverse properties

cos (x) = \frac{3}{5}

General solutions for

cos (x) = \frac{3}{5}

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

x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn

x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn

cos (x) = - \frac{3}{5} : x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

cos (x) = - \frac{3}{5}

Apply trig inverse properties

cos (x) = - \frac{3}{5}

General solutions for

cos (x) = - \frac{3}{5}

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

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

Combine all the solutions

x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn, x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

Show solutions in decimal form

x = 0.92729 \dots + 2 πn, x = 2 π - 0.92729 \dots + 2 πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

Graph

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