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Popular >

cos(20)+14sin^2(θ)=10

Solution

cos (2 0^{\circ}) + 14 sin^{2} (θ) = 10

Solution

θ = 0.93477 \dots + 36 0^{\circ} n, θ = 18 0^{\circ} - 0.93477 \dots + 36 0^{\circ} n, θ = - 0.93477 \dots + 36 0^{\circ} n, θ = 18 0^{\circ} + 0.93477 \dots + 36 0^{\circ} n

+1

Radians

θ = 0.93477 \dots + 2 πn, θ = π - 0.93477 \dots + 2 πn, θ = - 0.93477 \dots + 2 πn, θ = π + 0.93477 \dots + 2 πn

Solution steps

cos (2 0^{\circ}) + 14 sin^{2} (θ) = 10

Solve by substitution

cos (2 0^{\circ}) + 14 sin^{2} (θ) = 10

Let:

sin (θ) = u

cos (2 0^{\circ}) + 14 u^{2} = 10

cos (2 0^{\circ}) + 14 u^{2} = 10 : u = \frac{10 - cos ( 2 0 ^{\circ} )}{14}, u = - \frac{10 - cos ( 2 0 ^{\circ} )}{14}

cos (2 0^{\circ}) + 14 u^{2} = 10

Move

cos (2 0^{\circ})

to the right side

cos (2 0^{\circ}) + 14 u^{2} = 10

Subtract

cos (2 0^{\circ})

from both sides

cos (2 0^{\circ}) + 14 u^{2} - cos (2 0^{\circ}) = 10 - cos (2 0^{\circ})

Simplify

14 u^{2} = 10 - cos (2 0^{\circ})

14 u^{2} = 10 - cos (2 0^{\circ})

Divide both sides by

14

14 u^{2} = 10 - cos (2 0^{\circ})

Divide both sides by

14

\frac{14 u ^{2}}{14} = \frac{10}{14} - \frac{cos ( 2 0 ^{\circ} )}{14}

Simplify

\frac{14 u ^{2}}{14} = \frac{10}{14} - \frac{cos ( 2 0 ^{\circ} )}{14}

Simplify

\frac{14 u ^{2}}{14} : u^{2}

\frac{14 u ^{2}}{14}

Divide the numbers:

\frac{14}{14} = 1

= u^{2}

Simplify

\frac{10}{14} - \frac{cos ( 2 0 ^{\circ} )}{14} : \frac{10 - cos ( 2 0 ^{\circ} )}{14}

\frac{10}{14} - \frac{cos ( 2 0 ^{\circ} )}{14}

Apply rule

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

= \frac{10 - cos ( 2 0 ^{\circ} )}{14}

u^{2} = \frac{10 - cos ( 2 0 ^{\circ} )}{14}

u^{2} = \frac{10 - cos ( 2 0 ^{\circ} )}{14}

u^{2} = \frac{10 - cos ( 2 0 ^{\circ} )}{14}

For

x^{2} = f (a)

the solutions are

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

u = \frac{10 - cos ( 2 0 ^{\circ} )}{14}, u = - \frac{10 - cos ( 2 0 ^{\circ} )}{14}

Substitute back

u = sin (θ)

sin (θ) = \frac{10 - cos ( 2 0 ^{\circ} )}{14}, sin (θ) = - \frac{10 - cos ( 2 0 ^{\circ} )}{14}

sin (θ) = \frac{10 - cos ( 2 0 ^{\circ} )}{14}, sin (θ) = - \frac{10 - cos ( 2 0 ^{\circ} )}{14}

sin (θ) = \frac{10 - cos ( 2 0 ^{\circ} )}{14} : θ = arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = 18 0^{\circ} - arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n

sin (θ) = \frac{10 - cos ( 2 0 ^{\circ} )}{14}

Apply trig inverse properties

sin (θ) = \frac{10 - cos ( 2 0 ^{\circ} )}{14}

General solutions for

sin (θ) = \frac{10 - cos ( 2 0 ^{\circ} )}{14}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

θ = arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = 18 0^{\circ} - arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n

θ = arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = 18 0^{\circ} - arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n

sin (θ) = - \frac{10 - cos ( 2 0 ^{\circ} )}{14} : θ = arcsin (- \frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = 18 0^{\circ} + arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n

sin (θ) = - \frac{10 - cos ( 2 0 ^{\circ} )}{14}

Apply trig inverse properties

sin (θ) = - \frac{10 - cos ( 2 0 ^{\circ} )}{14}

General solutions for

sin (θ) = - \frac{10 - cos ( 2 0 ^{\circ} )}{14}

sin (x) = - a \Rightarrow x = arcsin (- a) + 36 0^{\circ} n, x = 18 0^{\circ} + arcsin (a) + 36 0^{\circ} n

θ = arcsin (- \frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = 18 0^{\circ} + arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n

θ = arcsin (- \frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = 18 0^{\circ} + arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n

Combine all the solutions

θ = arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = 18 0^{\circ} - arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = arcsin (- \frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n, θ = 18 0^{\circ} + arcsin (\frac{10 - cos ( 2 0 ^{\circ} )}{14}) + 36 0^{\circ} n

Show solutions in decimal form

θ = 0.93477 \dots + 36 0^{\circ} n, θ = 18 0^{\circ} - 0.93477 \dots + 36 0^{\circ} n, θ = - 0.93477 \dots + 36 0^{\circ} n, θ = 18 0^{\circ} + 0.93477 \dots + 36 0^{\circ} n

Graph

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