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Popular >

(6-4csc(θ))/(1-3csc(θ))<= 1-2sin(θ)

Solution

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} \leq 1 - 2 sin (θ)

Solution

False for all θ \in R

Solution steps

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} \leq 1 - 2 sin (θ)

Move

2 sin (θ)

to the left side

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} \leq 1 - 2 sin (θ)

Add

2 sin (θ)

to both sides

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} + 2 sin (θ) \leq 1 - 2 sin (θ) + 2 sin (θ)

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} + 2 sin (θ) \leq 1

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} + 2 sin (θ) \leq 1

Periodicity of

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} + 2 sin (θ) : 2 π

The compound periodicity of the sum of periodic functions is the least common multiplier of the periods

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )}, 2 sin (θ)

Periodicity of

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} : 2 π

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )}

is composed of the following functions and periods:

csc (θ)

with periodicity of

2 π

The compound periodicity is:

2 π

Periodicity of

2 sin (θ) : 2 π

Periodicity of

a \cdot sin (b x + c) + d = \frac{periodicity of sin ( x )}{∣ b ∣}

Periodicity of

sin (x)

2 π

= \frac{2 π}{∣ 1 ∣}

Simplify

= 2 π

Combine periods:

2 π, 2 π

= 2 π

Express with sin, cos

\frac{6 - 4 csc ( θ )}{1 - 3 csc ( θ )} + 2 sin (θ) \leq 1

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

\frac{6 - 4 \cdot \frac{1}{s i n ( θ )}}{1 - 3 \cdot \frac{1}{s i n ( θ )}} + 2 sin (θ) \leq 1

\frac{6 - 4 \cdot \frac{1}{s i n ( θ )}}{1 - 3 \cdot \frac{1}{s i n ( θ )}} + 2 sin (θ) \leq 1

Simplify

\frac{6 - 4 \cdot \frac{1}{s i n ( θ )}}{1 - 3 \cdot \frac{1}{s i n ( θ )}} + 2 sin (θ) : \frac{2 sin ^{2} ( θ ) - 4}{sin ( θ ) - 3}

\frac{6 - 4 \cdot \frac{1}{s i n ( θ )}}{1 - 3 \cdot \frac{1}{s i n ( θ )}} + 2 sin (θ)

\frac{6 - 4 \cdot \frac{1}{s i n ( θ )}}{1 - 3 \cdot \frac{1}{s i n ( θ )}} = \frac{6 sin ( θ ) - 4}{sin ( θ ) - 3}

\frac{6 - 4 \cdot \frac{1}{s i n ( θ )}}{1 - 3 \cdot \frac{1}{s i n ( θ )}}

3 \cdot \frac{1}{sin ( θ )} = \frac{3}{sin ( θ )}

3 \cdot \frac{1}{sin ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 3}{sin ( θ )}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{sin ( θ )}

= \frac{6 - 4 \cdot \frac{1}{s i n ( θ )}}{1 - \frac{3}{s i n ( θ )}}

4 \cdot \frac{1}{sin ( θ )} = \frac{4}{sin ( θ )}

4 \cdot \frac{1}{sin ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 4}{sin ( θ )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{sin ( θ )}

= \frac{6 - \frac{4}{s i n ( θ )}}{1 - \frac{3}{s i n ( θ )}}

Join

1 - \frac{3}{sin ( θ )} : \frac{sin ( θ ) - 3}{sin ( θ )}

1 - \frac{3}{sin ( θ )}

Convert element to fraction:

1 = \frac{1 sin ( θ )}{sin ( θ )}

= \frac{1 \cdot sin ( θ )}{sin ( θ )} - \frac{3}{sin ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot sin ( θ ) - 3}{sin ( θ )}

Multiply:

1 \cdot sin (θ) = sin (θ)

= \frac{sin ( θ ) - 3}{sin ( θ )}

= \frac{6 - \frac{4}{s i n ( θ )}}{\frac{s i n ( θ ) - 3}{s i n ( θ )}}

Join

6 - \frac{4}{sin ( θ )} : \frac{6 sin ( θ ) - 4}{sin ( θ )}

6 - \frac{4}{sin ( θ )}

Convert element to fraction:

6 = \frac{6 sin ( θ )}{sin ( θ )}

= \frac{6 sin ( θ )}{sin ( θ )} - \frac{4}{sin ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{6 sin ( θ ) - 4}{sin ( θ )}

= \frac{\frac{6 s i n ( θ ) - 4}{s i n ( θ )}}{\frac{s i n ( θ ) - 3}{s i n ( θ )}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( 6 sin ( θ ) - 4 ) sin ( θ )}{sin ( θ ) ( sin ( θ ) - 3 )}

Cancel the common factor:

sin (θ)

= \frac{6 sin ( θ ) - 4}{sin ( θ ) - 3}

= \frac{6 sin ( θ ) - 4}{sin ( θ ) - 3} + 2 sin (θ)

Convert element to fraction:

2 sin (θ) = \frac{2 sin ( θ ) ( sin ( θ ) - 3 )}{sin ( θ ) - 3}

= \frac{6 sin ( θ ) - 4}{sin ( θ ) - 3} + \frac{2 sin ( θ ) ( sin ( θ ) - 3 )}{sin ( θ ) - 3}

Since the denominators are equal, combine the fractions:

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

= \frac{6 sin ( θ ) - 4 + 2 sin ( θ ) ( sin ( θ ) - 3 )}{sin ( θ ) - 3}

Expand

6 sin (θ) - 4 + 2 sin (θ) (sin (θ) - 3) : 2 sin^{2} (θ) - 4

6 sin (θ) - 4 + 2 sin (θ) (sin (θ) - 3)

Expand

2 sin (θ) (sin (θ) - 3) : 2 sin^{2} (θ) - 6 sin (θ)

2 sin (θ) (sin (θ) - 3)

Apply the distributive law:

a (b - c) = ab - a c

a = 2 sin (θ), b = sin (θ), c = 3

= 2 sin (θ) sin (θ) - 2 sin (θ) \cdot 3

= 2 sin (θ) sin (θ) - 2 \cdot 3 sin (θ)

Simplify

2 sin (θ) sin (θ) - 2 \cdot 3 sin (θ) : 2 sin^{2} (θ) - 6 sin (θ)

2 sin (θ) sin (θ) - 2 \cdot 3 sin (θ)

2 sin (θ) sin (θ) = 2 sin^{2} (θ)

2 sin (θ) sin (θ)

Apply exponent rule:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= 2 sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= 2 sin^{2} (θ)

2 \cdot 3 sin (θ) = 6 sin (θ)

2 \cdot 3 sin (θ)

Multiply the numbers:

2 \cdot 3 = 6

= 6 sin (θ)

= 2 sin^{2} (θ) - 6 sin (θ)

= 2 sin^{2} (θ) - 6 sin (θ)

= 6 sin (θ) - 4 + 2 sin^{2} (θ) - 6 sin (θ)

Simplify

6 sin (θ) - 4 + 2 sin^{2} (θ) - 6 sin (θ) : 2 sin^{2} (θ) - 4

6 sin (θ) - 4 + 2 sin^{2} (θ) - 6 sin (θ)

Group like terms

= 6 sin (θ) + 2 sin^{2} (θ) - 6 sin (θ) - 4

Add similar elements:

6 sin (θ) - 6 sin (θ) = 0

= 2 sin^{2} (θ) - 4

= 2 sin^{2} (θ) - 4

= \frac{2 sin ^{2} ( θ ) - 4}{sin ( θ ) - 3}

= No Solution for θ \in R

False for all θ \in R

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