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Popular Trigonometry >

cos^2(t)+9/49 =1

Solution

cos^{2} (t) + \frac{9}{49} = 1

Solution

t = 0.44291 \dots + 2 πn, t = 2 π - 0.44291 \dots + 2 πn, t = 2.69868 \dots + 2 πn, t = - 2.69868 \dots + 2 πn

Degrees

t = 25.37693 \dots^{\circ} + 36 0^{\circ} n, t = 334.62306 \dots^{\circ} + 36 0^{\circ} n, t = 154.62306 \dots^{\circ} + 36 0^{\circ} n, t = - 154.62306 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (t) + \frac{9}{49} = 1

Solve by substitution

cos^{2} (t) + \frac{9}{49} = 1

Let:

cos (t) = u

u^{2} + \frac{9}{49} = 1

u^{2} + \frac{9}{49} = 1 : u = \frac{2 10}{7}, u = - \frac{2 10}{7}

u^{2} + \frac{9}{49} = 1

Move

\frac{9}{49}

to the right side

u^{2} + \frac{9}{49} = 1

Subtract

\frac{9}{49}

from both sides

u^{2} + \frac{9}{49} - \frac{9}{49} = 1 - \frac{9}{49}

Simplify

u^{2} = 1 - \frac{9}{49}

u^{2} = 1 - \frac{9}{49}

Simplify

1 - \frac{9}{49} : \frac{40}{49}

1 - \frac{9}{49}

Convert element to fraction:

1 = \frac{1 \cdot 49}{49}

= \frac{1 \cdot 49}{49} - \frac{9}{49}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 49 - 9}{49}

1 \cdot 49 - 9 = 40

1 \cdot 49 - 9

Multiply the numbers:

1 \cdot 49 = 49

= 49 - 9

Subtract the numbers:

49 - 9 = 40

= 40

= \frac{40}{49}

For

x^{2} = f (a)

the solutions are

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

u = \frac{40}{49}, u = - \frac{40}{49}

\frac{40}{49} = \frac{2 10}{7}

\frac{40}{49}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{40}{49}

49 = 7

49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

= \frac{40}{7}

40 = 210

40

Prime factorization of

40 : 2^{3} \cdot 5

40

40

divides by

240 = 20 \cdot 2

= 2 \cdot 20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 5

= 2^{3} \cdot 5

= 2^{3} \cdot 5

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2 \cdot 5

Apply radical rule:

= 2^{2} 2 \cdot 5

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 5

Refine

= 210

= \frac{2 10}{7}

- \frac{40}{49} = - \frac{2 10}{7}

- \frac{40}{49}

Simplify

\frac{40}{49} : \frac{2 10}{7}

\frac{40}{49}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{40}{49}

49 = 7

49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

= \frac{40}{7}

40 = 210

40

Prime factorization of

40 : 2^{3} \cdot 5

40

40

divides by

240 = 20 \cdot 2

= 2 \cdot 20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 5

= 2^{3} \cdot 5

= 2^{3} \cdot 5

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2 \cdot 5

Apply radical rule:

= 2^{2} 2 \cdot 5

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 5

Refine

= 210

= \frac{2 10}{7}

= - \frac{2 10}{7}

u = \frac{2 10}{7}, u = - \frac{2 10}{7}

Substitute back

u = cos (t)

cos (t) = \frac{2 10}{7}, cos (t) = - \frac{2 10}{7}

cos (t) = \frac{2 10}{7}, cos (t) = - \frac{2 10}{7}

cos (t) = \frac{2 10}{7} : t = arccos (\frac{2 10}{7}) + 2 πn, t = 2 π - arccos (\frac{2 10}{7}) + 2 πn

cos (t) = \frac{2 10}{7}

Apply trig inverse properties

cos (t) = \frac{2 10}{7}

General solutions for

cos (t) = \frac{2 10}{7}

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

t = arccos (\frac{2 10}{7}) + 2 πn, t = 2 π - arccos (\frac{2 10}{7}) + 2 πn

t = arccos (\frac{2 10}{7}) + 2 πn, t = 2 π - arccos (\frac{2 10}{7}) + 2 πn

cos (t) = - \frac{2 10}{7} : t = arccos (- \frac{2 10}{7}) + 2 πn, t = - arccos (- \frac{2 10}{7}) + 2 πn

cos (t) = - \frac{2 10}{7}

Apply trig inverse properties

cos (t) = - \frac{2 10}{7}

General solutions for

cos (t) = - \frac{2 10}{7}

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

t = arccos (- \frac{2 10}{7}) + 2 πn, t = - arccos (- \frac{2 10}{7}) + 2 πn

t = arccos (- \frac{2 10}{7}) + 2 πn, t = - arccos (- \frac{2 10}{7}) + 2 πn

Combine all the solutions

t = arccos (\frac{2 10}{7}) + 2 πn, t = 2 π - arccos (\frac{2 10}{7}) + 2 πn, t = arccos (- \frac{2 10}{7}) + 2 πn, t = - arccos (- \frac{2 10}{7}) + 2 πn

Show solutions in decimal form

t = 0.44291 \dots + 2 πn, t = 2 π - 0.44291 \dots + 2 πn, t = 2.69868 \dots + 2 πn, t = - 2.69868 \dots + 2 πn

Graph

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