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Popular Trigonometry >

tan(x)= 5/3 sin(x)[0.2pi]

Solution

tan (x) = \frac{5}{3} sin (x) [0.2 π]

Solution

x = 2 πn, x = π + 2 πn, x = 0.30137 \dots + 2 πn, x = 2 π - 0.30137 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 17.26743 \dots^{\circ} + 36 0^{\circ} n, x = 342.73256 \dots^{\circ} + 36 0^{\circ} n

Solution steps

tan (x) = \frac{5}{3} sin (x) [0.2 π]

Subtract

\frac{5}{3} sin (x) [0.2 π]

from both sides

tan (x) - \frac{π}{3} sin (x) = 0

Simplify

tan (x) - \frac{π}{3} sin (x) : \frac{3 tan ( x ) - π sin ( x )}{3}

tan (x) - \frac{π}{3} sin (x)

Multiply

\frac{π}{3} sin (x) : \frac{π sin ( x )}{3}

\frac{π}{3} sin (x)

Multiply fractions:

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

= \frac{π sin ( x )}{3}

= tan (x) - \frac{π sin ( x )}{3}

Convert element to fraction:

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

= \frac{tan ( x ) \cdot 3}{3} - \frac{π sin ( x )}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{tan ( x ) \cdot 3 - π sin ( x )}{3}

\frac{3 tan ( x ) - π sin ( x )}{3} = 0

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

3 tan (x) - π sin (x) = 0

Express with sin, cos

3 tan (x) - sin (x) π

Use the basic trigonometric identity:

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

= 3 \cdot \frac{sin ( x )}{cos ( x )} - sin (x) π

Simplify

3 \cdot \frac{sin ( x )}{cos ( x )} - sin (x) π : \frac{3 sin ( x ) - π sin ( x ) cos ( x )}{cos ( x )}

3 \cdot \frac{sin ( x )}{cos ( x )} - sin (x) π

Multiply

3 \cdot \frac{sin ( x )}{cos ( x )} : \frac{3 sin ( x )}{cos ( x )}

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

= \frac{sin ( x ) \cdot 3}{cos ( x )} - \frac{sin ( x ) π cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) \cdot 3 - sin ( x ) π cos ( x )}{cos ( x )}

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

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

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

3 sin (x) - cos (x) sin (x) π = 0

Factor

3 sin (x) - cos (x) sin (x) π : - sin (x) (π cos (x) - 3)

3 sin (x) - cos (x) sin (x) π

Factor out common term

- sin (x)

= - sin (x) (- 3 + π cos (x))

- sin (x) (π cos (x) - 3) = 0

Solving each part separately

sin (x) = 0 or π cos (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

π cos (x) - 3 = 0 : x = arccos (\frac{3}{π}) + 2 πn, x = 2 π - arccos (\frac{3}{π}) + 2 πn

π cos (x) - 3 = 0

Move

3

to the right side

π cos (x) - 3 = 0

Add

3

to both sides

π cos (x) - 3 + 3 = 0 + 3

Simplify

π cos (x) = 3

π cos (x) = 3

Divide both sides by

π

π cos (x) = 3

Divide both sides by

π

\frac{π cos ( x )}{π} = \frac{3}{π}

Simplify

cos (x) = \frac{3}{π}

cos (x) = \frac{3}{π}

Apply trig inverse properties

cos (x) = \frac{3}{π}

General solutions for

cos (x) = \frac{3}{π}

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.30137 \dots + 2 πn, x = 2 π - 0.30137 \dots + 2 πn

Graph

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