What is the general solution for 8cos(x)=3csc(x) ?

The general solution for 8cos(x)=3csc(x) is x=(0.84806…)/2+pin,x= pi/2-(0.84806…)/2+pin

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8cos(x)=3csc(x)

Solution

8 cos (x) = 3 csc (x)

Solution

x = \frac{0.84806 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.84806 \dots}{2} + πn

Degrees

x = 24.29518 \dots^{\circ} + 18 0^{\circ} n, x = 65.70481 \dots^{\circ} + 18 0^{\circ} n

Solution steps

8 cos (x) = 3 csc (x)

Subtract

3 csc (x)

from both sides

8 cos (x) - 3 csc (x) = 0

Express with sin, cos

- 3 csc (x) + 8 cos (x)

Use the basic trigonometric identity:

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

= - 3 \cdot \frac{1}{sin ( x )} + 8 cos (x)

Simplify

- 3 \cdot \frac{1}{sin ( x )} + 8 cos (x) : \frac{- 3 + 8 cos ( x ) sin ( x )}{sin ( x )}

- 3 \cdot \frac{1}{sin ( x )} + 8 cos (x)

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

3 \cdot \frac{1}{sin ( x )}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{sin ( x )}

= - \frac{3}{sin ( x )} + 8 cos (x)

Convert element to fraction:

8 cos (x) = \frac{8 cos ( x ) sin ( x )}{sin ( x )}

= - \frac{3}{sin ( x )} + \frac{8 cos ( x ) sin ( x )}{sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 3 + 8 cos ( x ) sin ( x )}{sin ( x )}

= \frac{- 3 + 8 cos ( x ) sin ( x )}{sin ( x )}

\frac{- 3 + 8 cos ( x ) sin ( x )}{sin ( x )} = 0

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

- 3 + 8 cos (x) sin (x) = 0

Rewrite using trig identities

- 3 + 8 cos (x) sin (x)

Use the Double Angle identity:

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

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

= - 3 + 8 \cdot \frac{sin ( 2 x )}{2}

- 3 + 8 \cdot \frac{sin ( 2 x )}{2} = 0

8 \cdot \frac{sin ( 2 x )}{2} = 4 sin (2 x)

8 \cdot \frac{sin ( 2 x )}{2}

Multiply fractions:

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

= \frac{sin ( 2 x ) \cdot 8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 sin (2 x)

- 3 + 4 sin (2 x) = 0

Move

3

to the right side

- 3 + 4 sin (2 x) = 0

Add

3

to both sides

- 3 + 4 sin (2 x) + 3 = 0 + 3

Simplify

4 sin (2 x) = 3

4 sin (2 x) = 3

Divide both sides by

4

4 sin (2 x) = 3

Divide both sides by

4

\frac{4 sin ( 2 x )}{4} = \frac{3}{4}

Simplify

sin (2 x) = \frac{3}{4}

sin (2 x) = \frac{3}{4}

Apply trig inverse properties

sin (2 x) = \frac{3}{4}

General solutions for

sin (2 x) = \frac{3}{4}

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

2 x = arcsin (\frac{3}{4}) + 2 πn, 2 x = π - arcsin (\frac{3}{4}) + 2 πn

2 x = arcsin (\frac{3}{4}) + 2 πn, 2 x = π - arcsin (\frac{3}{4}) + 2 πn

Solve

2 x = arcsin (\frac{3}{4}) + 2 πn : x = \frac{arcsin ( \frac{3}{4} )}{2} + πn

2 x = arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

2

2 x = arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsin ( \frac{3}{4} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arcsin ( \frac{3}{4} )}{2} + πn

x = \frac{arcsin ( \frac{3}{4} )}{2} + πn

Solve

2 x = π - arcsin (\frac{3}{4}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{3}{4} )}{2} + πn

2 x = π - arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

2

2 x = π - arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{3}{4} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} - \frac{arcsin ( \frac{3}{4} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{3}{4} )}{2} + πn

x = \frac{arcsin ( \frac{3}{4} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{3}{4} )}{2} + πn

Show solutions in decimal form

x = \frac{0.84806 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.84806 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 8cos(x)=3csc(x) ?
The general solution for 8cos(x)=3csc(x) is x=(0.84806…)/2+pin,x= pi/2-(0.84806…)/2+pin