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Popular Trigonometry >

36/49+cos^2(θ)=1

Solution

\frac{36}{49} + cos^{2} (θ) = 1

Solution

θ = 1.02969 \dots + 2 πn, θ = 2 π - 1.02969 \dots + 2 πn, θ = 2.11189 \dots + 2 πn, θ = - 2.11189 \dots + 2 πn

Degrees

θ = 58.99728 \dots^{\circ} + 36 0^{\circ} n, θ = 301.00271 \dots^{\circ} + 36 0^{\circ} n, θ = 121.00271 \dots^{\circ} + 36 0^{\circ} n, θ = - 121.00271 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{36}{49} + cos^{2} (θ) = 1

Solve by substitution

\frac{36}{49} + cos^{2} (θ) = 1

Let:

cos (θ) = u

\frac{36}{49} + u^{2} = 1

\frac{36}{49} + u^{2} = 1 : u = \frac{13}{7}, u = - \frac{13}{7}

\frac{36}{49} + u^{2} = 1

Move

\frac{36}{49}

to the right side

\frac{36}{49} + u^{2} = 1

Subtract

\frac{36}{49}

from both sides

\frac{36}{49} + u^{2} - \frac{36}{49} = 1 - \frac{36}{49}

Simplify

u^{2} = 1 - \frac{36}{49}

u^{2} = 1 - \frac{36}{49}

Simplify

1 - \frac{36}{49} : \frac{13}{49}

1 - \frac{36}{49}

Convert element to fraction:

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

= \frac{1 \cdot 49}{49} - \frac{36}{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 - 36}{49}

1 \cdot 49 - 36 = 13

1 \cdot 49 - 36

Multiply the numbers:

1 \cdot 49 = 49

= 49 - 36

Subtract the numbers:

49 - 36 = 13

= 13

= \frac{13}{49}

For

x^{2} = f (a)

the solutions are

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

u = \frac{13}{49}, u = - \frac{13}{49}

\frac{13}{49} = \frac{13}{7}

\frac{13}{49}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{13}{49}

49 = 7

49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

= \frac{13}{7}

- \frac{13}{49} = - \frac{13}{7}

- \frac{13}{49}

Simplify

\frac{13}{49} : \frac{13}{7}

\frac{13}{49}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{13}{49}

49 = 7

49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

= \frac{13}{7}

= - \frac{13}{7}

u = \frac{13}{7}, u = - \frac{13}{7}

Substitute back

u = cos (θ)

cos (θ) = \frac{13}{7}, cos (θ) = - \frac{13}{7}

cos (θ) = \frac{13}{7}, cos (θ) = - \frac{13}{7}

cos (θ) = \frac{13}{7} : θ = arccos (\frac{13}{7}) + 2 πn, θ = 2 π - arccos (\frac{13}{7}) + 2 πn

cos (θ) = \frac{13}{7}

Apply trig inverse properties

cos (θ) = \frac{13}{7}

General solutions for

cos (θ) = \frac{13}{7}

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

θ = arccos (\frac{13}{7}) + 2 πn, θ = 2 π - arccos (\frac{13}{7}) + 2 πn

θ = arccos (\frac{13}{7}) + 2 πn, θ = 2 π - arccos (\frac{13}{7}) + 2 πn

cos (θ) = - \frac{13}{7} : θ = arccos (- \frac{13}{7}) + 2 πn, θ = - arccos (- \frac{13}{7}) + 2 πn

cos (θ) = - \frac{13}{7}

Apply trig inverse properties

cos (θ) = - \frac{13}{7}

General solutions for

cos (θ) = - \frac{13}{7}

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

θ = arccos (- \frac{13}{7}) + 2 πn, θ = - arccos (- \frac{13}{7}) + 2 πn

θ = arccos (- \frac{13}{7}) + 2 πn, θ = - arccos (- \frac{13}{7}) + 2 πn

Combine all the solutions

θ = arccos (\frac{13}{7}) + 2 πn, θ = 2 π - arccos (\frac{13}{7}) + 2 πn, θ = arccos (- \frac{13}{7}) + 2 πn, θ = - arccos (- \frac{13}{7}) + 2 πn

Show solutions in decimal form

θ = 1.02969 \dots + 2 πn, θ = 2 π - 1.02969 \dots + 2 πn, θ = 2.11189 \dots + 2 πn, θ = - 2.11189 \dots + 2 πn

Graph

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