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Popular Trigonometry >

1/36+cos^2(θ)=1

Solution

\frac{1}{36} + cos^{2} (θ) = 1

Solution

θ = 0.16744 \dots + 2 πn, θ = 2 π - 0.16744 \dots + 2 πn, θ = 2.97414 \dots + 2 πn, θ = - 2.97414 \dots + 2 πn

Degrees

θ = 9.59406 \dots^{\circ} + 36 0^{\circ} n, θ = 350.40593 \dots^{\circ} + 36 0^{\circ} n, θ = 170.40593 \dots^{\circ} + 36 0^{\circ} n, θ = - 170.40593 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{36} + cos^{2} (θ) = 1

Solve by substitution

\frac{1}{36} + cos^{2} (θ) = 1

Let:

cos (θ) = u

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

\frac{1}{36} + u^{2} = 1 : u = \frac{35}{6}, u = - \frac{35}{6}

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

Move

\frac{1}{36}

to the right side

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

Subtract

\frac{1}{36}

from both sides

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

Simplify

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

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

Simplify

1 - \frac{1}{36} : \frac{35}{36}

1 - \frac{1}{36}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 36 - 1 = 35

1 \cdot 36 - 1

Multiply the numbers:

1 \cdot 36 = 36

= 36 - 1

Subtract the numbers:

36 - 1 = 35

= 35

= \frac{35}{36}

For

x^{2} = f (a)

the solutions are

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

u = \frac{35}{36}, u = - \frac{35}{36}

\frac{35}{36} = \frac{35}{6}

\frac{35}{36}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{35}{36}

36 = 6

36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

= \frac{35}{6}

- \frac{35}{36} = - \frac{35}{6}

- \frac{35}{36}

Simplify

\frac{35}{36} : \frac{35}{6}

\frac{35}{36}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{35}{36}

36 = 6

36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

= \frac{35}{6}

= - \frac{35}{6}

u = \frac{35}{6}, u = - \frac{35}{6}

Substitute back

u = cos (θ)

cos (θ) = \frac{35}{6}, cos (θ) = - \frac{35}{6}

cos (θ) = \frac{35}{6}, cos (θ) = - \frac{35}{6}

cos (θ) = \frac{35}{6} : θ = arccos (\frac{35}{6}) + 2 πn, θ = 2 π - arccos (\frac{35}{6}) + 2 πn

cos (θ) = \frac{35}{6}

Apply trig inverse properties

cos (θ) = \frac{35}{6}

General solutions for

cos (θ) = \frac{35}{6}

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

θ = arccos (\frac{35}{6}) + 2 πn, θ = 2 π - arccos (\frac{35}{6}) + 2 πn

θ = arccos (\frac{35}{6}) + 2 πn, θ = 2 π - arccos (\frac{35}{6}) + 2 πn

cos (θ) = - \frac{35}{6} : θ = arccos (- \frac{35}{6}) + 2 πn, θ = - arccos (- \frac{35}{6}) + 2 πn

cos (θ) = - \frac{35}{6}

Apply trig inverse properties

cos (θ) = - \frac{35}{6}

General solutions for

cos (θ) = - \frac{35}{6}

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

θ = arccos (- \frac{35}{6}) + 2 πn, θ = - arccos (- \frac{35}{6}) + 2 πn

θ = arccos (- \frac{35}{6}) + 2 πn, θ = - arccos (- \frac{35}{6}) + 2 πn

Combine all the solutions

θ = arccos (\frac{35}{6}) + 2 πn, θ = 2 π - arccos (\frac{35}{6}) + 2 πn, θ = arccos (- \frac{35}{6}) + 2 πn, θ = - arccos (- \frac{35}{6}) + 2 πn

Show solutions in decimal form

θ = 0.16744 \dots + 2 πn, θ = 2 π - 0.16744 \dots + 2 πn, θ = 2.97414 \dots + 2 πn, θ = - 2.97414 \dots + 2 πn

Graph

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