What is the general solution for sin(θ)=(0.3924)/(cos(θ)) ?

The general solution for sin(θ)=(0.3924)/(cos(θ)) is θ=(0.90237…)/2+pin,θ= pi/2-(0.90237…)/2+pin

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sin(θ)=(0.3924)/(cos(θ))

Solution

sin (θ) = \frac{0.3924}{cos ( θ )}

Solution

θ = \frac{0.90237 \dots}{2} + πn, θ = \frac{π}{2} - \frac{0.90237 \dots}{2} + πn

Degrees

θ = 25.85109 \dots^{\circ} + 18 0^{\circ} n, θ = 64.14890 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (θ) = \frac{0.3924}{cos ( θ )}

Subtract

\frac{0.3924}{cos ( θ )}

from both sides

sin (θ) - \frac{0.3924}{cos ( θ )} = 0

Simplify

sin (θ) - \frac{0.3924}{cos ( θ )} : \frac{sin ( θ ) cos ( θ ) - 0.3924}{cos ( θ )}

sin (θ) - \frac{0.3924}{cos ( θ )}

Convert element to fraction:

sin (θ) = \frac{sin ( θ ) cos ( θ )}{cos ( θ )}

= \frac{sin ( θ ) cos ( θ )}{cos ( θ )} - \frac{0.3924}{cos ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( θ ) cos ( θ ) - 0.3924}{cos ( θ )}

\frac{sin ( θ ) cos ( θ ) - 0.3924}{cos ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (θ) cos (θ) - 0.3924 = 0

Rewrite using trig identities

sin (θ) cos (θ) - 0.3924

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= - 0.3924 + \frac{sin ( 2 θ )}{2}

- 0.3924 + \frac{sin ( 2 θ )}{2} = 0

Move

0.3924

to the right side

- 0.3924 + \frac{sin ( 2 θ )}{2} = 0

Add

0.3924

to both sides

- 0.3924 + \frac{sin ( 2 θ )}{2} + 0.3924 = 0 + 0.3924

Simplify

\frac{sin ( 2 θ )}{2} = 0.3924

\frac{sin ( 2 θ )}{2} = 0.3924

Multiply both sides by

2

\frac{sin ( 2 θ )}{2} = 0.3924

Multiply both sides by

2

\frac{2 sin ( 2 θ )}{2} = 0.3924 \cdot 2

Simplify

sin (2 θ) = 0.7848

sin (2 θ) = 0.7848

Apply trig inverse properties

sin (2 θ) = 0.7848

General solutions for

sin (2 θ) = 0.7848

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

2 θ = arcsin (0.7848) + 2 πn, 2 θ = π - arcsin (0.7848) + 2 πn

2 θ = arcsin (0.7848) + 2 πn, 2 θ = π - arcsin (0.7848) + 2 πn

Solve

2 θ = arcsin (0.7848) + 2 πn : θ = \frac{arcsin ( 0.7848 )}{2} + πn

2 θ = arcsin (0.7848) + 2 πn

Divide both sides by

2

2 θ = arcsin (0.7848) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{arcsin ( 0.7848 )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{arcsin ( 0.7848 )}{2} + πn

θ = \frac{arcsin ( 0.7848 )}{2} + πn

Solve

2 θ = π - arcsin (0.7848) + 2 πn : θ = \frac{π}{2} - \frac{arcsin ( 0.7848 )}{2} + πn

2 θ = π - arcsin (0.7848) + 2 πn

Divide both sides by

2

2 θ = π - arcsin (0.7848) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} - \frac{arcsin ( 0.7848 )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} - \frac{arcsin ( 0.7848 )}{2} + πn

θ = \frac{π}{2} - \frac{arcsin ( 0.7848 )}{2} + πn

θ = \frac{arcsin ( 0.7848 )}{2} + πn, θ = \frac{π}{2} - \frac{arcsin ( 0.7848 )}{2} + πn

Show solutions in decimal form

θ = \frac{0.90237 \dots}{2} + πn, θ = \frac{π}{2} - \frac{0.90237 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)=(0.3924)/(cos(θ)) ?
The general solution for sin(θ)=(0.3924)/(cos(θ)) is θ=(0.90237…)/2+pin,θ= pi/2-(0.90237…)/2+pin