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Popular Trigonometry >

9/73+sin^2(θ)=1

Solution

\frac{9}{73} + sin^{2} (θ) = 1

Solution

θ = 1.21202 \dots + 2 πn, θ = π - 1.21202 \dots + 2 πn, θ = - 1.21202 \dots + 2 πn, θ = π + 1.21202 \dots + 2 πn

Degrees

θ = 69.44395 \dots^{\circ} + 36 0^{\circ} n, θ = 110.55604 \dots^{\circ} + 36 0^{\circ} n, θ = - 69.44395 \dots^{\circ} + 36 0^{\circ} n, θ = 249.44395 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{9}{73} + sin^{2} (θ) = 1

Solve by substitution

\frac{9}{73} + sin^{2} (θ) = 1

Let:

sin (θ) = u

\frac{9}{73} + u^{2} = 1

\frac{9}{73} + u^{2} = 1 : u = \frac{8 73}{73}, u = - \frac{8 73}{73}

\frac{9}{73} + u^{2} = 1

Move

\frac{9}{73}

to the right side

\frac{9}{73} + u^{2} = 1

Subtract

\frac{9}{73}

from both sides

\frac{9}{73} + u^{2} - \frac{9}{73} = 1 - \frac{9}{73}

Simplify

u^{2} = 1 - \frac{9}{73}

u^{2} = 1 - \frac{9}{73}

Simplify

1 - \frac{9}{73} : \frac{64}{73}

1 - \frac{9}{73}

Convert element to fraction:

1 = \frac{1 \cdot 73}{73}

= \frac{1 \cdot 73}{73} - \frac{9}{73}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 73 - 9}{73}

1 \cdot 73 - 9 = 64

1 \cdot 73 - 9

Multiply the numbers:

1 \cdot 73 = 73

= 73 - 9

Subtract the numbers:

73 - 9 = 64

= 64

= \frac{64}{73}

For

x^{2} = f (a)

the solutions are

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

u = \frac{64}{73}, u = - \frac{64}{73}

\frac{64}{73} = \frac{8 73}{73}

\frac{64}{73}

Apply radical rule:

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

assuming

a \geq 0, b \geq 0

= \frac{64}{73}

64 = 8

64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

n a^{n} = a

8^{2} = 8

= 8

= \frac{8}{73}

Rationalize

\frac{8}{73} : \frac{8 73}{73}

\frac{8}{73}

Multiply by the conjugate

\frac{73}{73}

= \frac{8 73}{73 73}

7373 = 73

7373

Apply radical rule:

a a = a

7373 = 73

= 73

= \frac{8 73}{73}

= \frac{8 73}{73}

- \frac{64}{73} = - \frac{8 73}{73}

- \frac{64}{73}

Simplify

\frac{64}{73} : \frac{8}{73}

\frac{64}{73}

Apply radical rule:

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

assuming

a \geq 0, b \geq 0

= \frac{64}{73}

64 = 8

64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

n a^{n} = a

8^{2} = 8

= 8

= \frac{8}{73}

= - \frac{8}{73}

Rationalize

- \frac{8}{73} : - \frac{8 73}{73}

- \frac{8}{73}

Multiply by the conjugate

\frac{73}{73}

= - \frac{8 73}{73 73}

7373 = 73

7373

Apply radical rule:

a a = a

7373 = 73

= 73

= - \frac{8 73}{73}

= - \frac{8 73}{73}

u = \frac{8 73}{73}, u = - \frac{8 73}{73}

Substitute back

u = sin (θ)

sin (θ) = \frac{8 73}{73}, sin (θ) = - \frac{8 73}{73}

sin (θ) = \frac{8 73}{73}, sin (θ) = - \frac{8 73}{73}

sin (θ) = \frac{8 73}{73} : θ = arcsin (\frac{8 73}{73}) + 2 πn, θ = π - arcsin (\frac{8 73}{73}) + 2 πn

sin (θ) = \frac{8 73}{73}

Apply trig inverse properties

sin (θ) = \frac{8 73}{73}

General solutions for

sin (θ) = \frac{8 73}{73}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

θ = arcsin (\frac{8 73}{73}) + 2 πn, θ = π - arcsin (\frac{8 73}{73}) + 2 πn

θ = arcsin (\frac{8 73}{73}) + 2 πn, θ = π - arcsin (\frac{8 73}{73}) + 2 πn

sin (θ) = - \frac{8 73}{73} : θ = arcsin (- \frac{8 73}{73}) + 2 πn, θ = π + arcsin (\frac{8 73}{73}) + 2 πn

sin (θ) = - \frac{8 73}{73}

Apply trig inverse properties

sin (θ) = - \frac{8 73}{73}

General solutions for

sin (θ) = - \frac{8 73}{73}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

θ = arcsin (- \frac{8 73}{73}) + 2 πn, θ = π + arcsin (\frac{8 73}{73}) + 2 πn

θ = arcsin (- \frac{8 73}{73}) + 2 πn, θ = π + arcsin (\frac{8 73}{73}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{8 73}{73}) + 2 πn, θ = π - arcsin (\frac{8 73}{73}) + 2 πn, θ = arcsin (- \frac{8 73}{73}) + 2 πn, θ = π + arcsin (\frac{8 73}{73}) + 2 πn

Show solutions in decimal form

θ = 1.21202 \dots + 2 πn, θ = π - 1.21202 \dots + 2 πn, θ = - 1.21202 \dots + 2 πn, θ = π + 1.21202 \dots + 2 πn

Graph

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