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

The general solution for sin(θ)=(tan(θ))/(sin(θ)) is No Solution for θ\in\mathbb{R}

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sin(θ)=(tan(θ))/(sin(θ))

Solution

sin (θ) = \frac{tan ( θ )}{sin ( θ )}

Solution

No Solution for θ \in R

Solution steps

sin (θ) = \frac{tan ( θ )}{sin ( θ )}

Subtract

\frac{tan ( θ )}{sin ( θ )}

from both sides

sin (θ) - \frac{tan ( θ )}{sin ( θ )} = 0

Simplify

sin (θ) - \frac{tan ( θ )}{sin ( θ )} : \frac{sin ^{2} ( θ ) - tan ( θ )}{sin ( θ )}

sin (θ) - \frac{tan ( θ )}{sin ( θ )}

Convert element to fraction:

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

= \frac{sin ( θ ) sin ( θ )}{sin ( θ )} - \frac{tan ( θ )}{sin ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( θ ) sin ( θ ) - tan ( θ )}{sin ( θ )}

sin (θ) sin (θ) - tan (θ) = sin^{2} (θ) - tan (θ)

sin (θ) sin (θ) - tan (θ)

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= sin^{2} (θ)

= sin^{2} (θ) - tan (θ)

= \frac{sin ^{2} ( θ ) - tan ( θ )}{sin ( θ )}

\frac{sin ^{2} ( θ ) - tan ( θ )}{sin ( θ )} = 0

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

sin^{2} (θ) - tan (θ) = 0

Express with sin, cos

sin^{2} (θ) - tan (θ)

Use the basic trigonometric identity:

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

= sin^{2} (θ) - \frac{sin ( θ )}{cos ( θ )}

Simplify

sin^{2} (θ) - \frac{sin ( θ )}{cos ( θ )} : \frac{sin ^{2} ( θ ) cos ( θ ) - sin ( θ )}{cos ( θ )}

sin^{2} (θ) - \frac{sin ( θ )}{cos ( θ )}

Convert element to fraction:

sin^{2} (θ) = \frac{sin ^{2} ( θ ) cos ( θ )}{cos ( θ )}

= \frac{sin ^{2} ( θ ) cos ( θ )}{cos ( θ )} - \frac{sin ( θ )}{cos ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( θ ) cos ( θ ) - sin ( θ )}{cos ( θ )}

= \frac{sin ^{2} ( θ ) cos ( θ ) - sin ( θ )}{cos ( θ )}

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

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

- sin (θ) + cos (θ) sin^{2} (θ) = 0

Factor

- sin (θ) + cos (θ) sin^{2} (θ) : sin (θ) (- 1 + sin (θ) cos (θ))

- sin (θ) + cos (θ) sin^{2} (θ)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{2} (θ) cos (θ) = sin (θ) sin (θ)

= - sin (θ) + sin (θ) sin (θ)

Factor out common term

sin (θ)

= sin (θ) (- 1 + sin (θ) cos (θ))

sin (θ) (- 1 + sin (θ) cos (θ)) = 0

Solving each part separately

sin (θ) = 0 or - 1 + sin (θ) cos (θ) = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

- 1 + sin (θ) cos (θ) = 0 :

No Solution

- 1 + sin (θ) cos (θ) = 0

Rewrite using trig identities

- 1 + sin (θ) cos (θ)

Use the Double Angle identity:

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

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

= - 1 + \frac{sin ( 2 θ )}{2}

- 1 + \frac{sin ( 2 θ )}{2} = 0

Move

1

to the right side

- 1 + \frac{sin ( 2 θ )}{2} = 0

Add

1

to both sides

- 1 + \frac{sin ( 2 θ )}{2} + 1 = 0 + 1

Simplify

\frac{sin ( 2 θ )}{2} = 1

\frac{sin ( 2 θ )}{2} = 1

Multiply both sides by

2

\frac{sin ( 2 θ )}{2} = 1

Multiply both sides by

2

\frac{2 sin ( 2 θ )}{2} = 1 \cdot 2

Simplify

sin (2 θ) = 2

sin (2 θ) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for θ \in R

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24=32+8-2*sqrt(32*8)*cos(θ)

24 = 32 + 8 - 2 \cdot 32 \cdot 8 \cdot cos (θ)

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tan(x)*sin(x)=5219

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)=(tan(θ))/(sin(θ)) ?
The general solution for sin(θ)=(tan(θ))/(sin(θ)) is No Solution for θ\in\mathbb{R}