What is the general solution for 7cot(θ)=3csc(θ) ?

The general solution for 7cot(θ)=3csc(θ) is θ=1.12788…+2pin,θ=2pi-1.12788…+2pin

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Popular >

7cot(θ)=3csc(θ)

Solution

7 cot (θ) = 3 csc (θ)

Solution

θ = 1.12788 \dots + 2 πn, θ = 2 π - 1.12788 \dots + 2 πn

Degrees

θ = 64.62306 \dots^{\circ} + 36 0^{\circ} n, θ = 295.37693 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 cot (θ) = 3 csc (θ)

Subtract

3 csc (θ)

from both sides

7 cot (θ) - 3 csc (θ) = 0

Express with sin, cos

- 3 csc (θ) + 7 cot (θ)

Use the basic trigonometric identity:

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

= - 3 \cdot \frac{1}{sin ( θ )} + 7 cot (θ)

Use the basic trigonometric identity:

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

= - 3 \cdot \frac{1}{sin ( θ )} + 7 \cdot \frac{cos ( θ )}{sin ( θ )}

Simplify

- 3 \cdot \frac{1}{sin ( θ )} + 7 \cdot \frac{cos ( θ )}{sin ( θ )} : \frac{- 3 + 7 cos ( θ )}{sin ( θ )}

- 3 \cdot \frac{1}{sin ( θ )} + 7 \cdot \frac{cos ( θ )}{sin ( 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 )}

7 \cdot \frac{cos ( θ )}{sin ( θ )} = \frac{7 cos ( θ )}{sin ( θ )}

7 \cdot \frac{cos ( θ )}{sin ( θ )}

Multiply fractions:

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

= \frac{cos ( θ ) \cdot 7}{sin ( θ )}

= - \frac{3}{sin ( θ )} + \frac{7 cos ( θ )}{sin ( θ )}

Apply rule

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

= \frac{- 3 + 7 cos ( θ )}{sin ( θ )}

= \frac{- 3 + 7 cos ( θ )}{sin ( θ )}

\frac{- 3 + 7 cos ( θ )}{sin ( θ )} = 0

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

- 3 + 7 cos (θ) = 0

Move

3

to the right side

- 3 + 7 cos (θ) = 0

Add

3

to both sides

- 3 + 7 cos (θ) + 3 = 0 + 3

Simplify

7 cos (θ) = 3

7 cos (θ) = 3

Divide both sides by

7

7 cos (θ) = 3

Divide both sides by

7

\frac{7 cos ( θ )}{7} = \frac{3}{7}

Simplify

cos (θ) = \frac{3}{7}

cos (θ) = \frac{3}{7}

Apply trig inverse properties

cos (θ) = \frac{3}{7}

General solutions for

cos (θ) = \frac{3}{7}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

θ = arccos (\frac{3}{7}) + 2 πn, θ = 2 π - arccos (\frac{3}{7}) + 2 πn

θ = arccos (\frac{3}{7}) + 2 πn, θ = 2 π - arccos (\frac{3}{7}) + 2 πn

Show solutions in decimal form

θ = 1.12788 \dots + 2 πn, θ = 2 π - 1.12788 \dots + 2 πn

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sin (3 t) = 0

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sin^{2} (x) - 3 cos (x) - 3 = 0

csc^2(x)=0

csc^{2} (x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for 7cot(θ)=3csc(θ) ?
The general solution for 7cot(θ)=3csc(θ) is θ=1.12788…+2pin,θ=2pi-1.12788…+2pin