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Popular Trigonometry >

tan^2(x)=5sin^2(x)

Solution

tan^{2} (x) = 5 sin^{2} (x)

Solution

x = 2 πn, x = π + 2 πn, x = 2.03444 \dots + 2 πn, x = - 2.03444 \dots + 2 πn, x = 1.10714 \dots + 2 πn, x = 2 π - 1.10714 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 116.56505 \dots^{\circ} + 36 0^{\circ} n, x = - 116.56505 \dots^{\circ} + 36 0^{\circ} n, x = 63.43494 \dots^{\circ} + 36 0^{\circ} n, x = 296.56505 \dots^{\circ} + 36 0^{\circ} n

Solution steps

tan^{2} (x) = 5 sin^{2} (x)

Subtract

5 sin^{2} (x)

from both sides

tan^{2} (x) - 5 sin^{2} (x) = 0

Factor

tan^{2} (x) - 5 sin^{2} (x) : (tan (x) + 5 sin (x)) (tan (x) - 5 sin (x))

tan^{2} (x) - 5 sin^{2} (x)

Rewrite

tan^{2} (x) - 5 sin^{2} (x)

tan^{2} (x) - (5 sin (x))^{2}

tan^{2} (x) - 5 sin^{2} (x)

Apply radical rule:

a = (a)^{2}

5 = (5)^{2}

= tan^{2} (x) - (5)^{2} sin^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(5)^{2} sin^{2} (x) = (5 sin (x))^{2}

= tan^{2} (x) - (5 sin (x))^{2}

= tan^{2} (x) - (5 sin (x))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

tan^{2} (x) - (5 sin (x))^{2} = (tan (x) + 5 sin (x)) (tan (x) - 5 sin (x))

= (tan (x) + 5 sin (x)) (tan (x) - 5 sin (x))

(tan (x) + 5 sin (x)) (tan (x) - 5 sin (x)) = 0

Solving each part separately

tan (x) + 5 sin (x) = 0 or tan (x) - 5 sin (x) = 0

tan (x) + 5 sin (x) = 0 : x = 2 πn, x = π + 2 πn, x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

tan (x) + 5 sin (x) = 0

Express with sin, cos

tan (x) + sin (x) 5

Use the basic trigonometric identity:

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

= \frac{sin ( x )}{cos ( x )} + sin (x) 5

Simplify

\frac{sin ( x )}{cos ( x )} + sin (x) 5 : \frac{sin ( x ) + 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x )}{cos ( x )} + sin (x) 5

Convert element to fraction:

5 sin (x) = \frac{sin ( x ) 5 cos ( x )}{cos ( x )}

= \frac{sin ( x )}{cos ( x )} + \frac{sin ( x ) 5 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 ) + sin ( x ) 5 cos ( x )}{cos ( x )}

= \frac{sin ( x ) + 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x ) + cos ( x ) sin ( x ) 5}{cos ( x )} = 0

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

sin (x) + cos (x) sin (x) 5 = 0

Factor

sin (x) + cos (x) sin (x) 5 : sin (x) (1 + 5 cos (x))

sin (x) + cos (x) sin (x) 5

Factor out common term

sin (x)

= sin (x) (1 + cos (x) 5)

sin (x) (1 + 5 cos (x)) = 0

Solving each part separately

sin (x) = 0 or 1 + 5 cos (x) = 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

1 + 5 cos (x) = 0 : x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

1 + 5 cos (x) = 0

Move

1

to the right side

1 + 5 cos (x) = 0

Subtract

1

from both sides

1 + 5 cos (x) - 1 = 0 - 1

Simplify

5 cos (x) = - 1

5 cos (x) = - 1

Divide both sides by

5

5 cos (x) = - 1

Divide both sides by

5

\frac{5 cos ( x )}{5} = \frac{- 1}{5}

Simplify

\frac{5 cos ( x )}{5} = \frac{- 1}{5}

Simplify

\frac{5 cos ( x )}{5} : cos (x)

\frac{5 cos ( x )}{5}

Cancel the common factor:

5

= cos (x)

Simplify

\frac{- 1}{5} : - \frac{5}{5}

\frac{- 1}{5}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{5}

Rationalize

- \frac{1}{5} : - \frac{5}{5}

- \frac{1}{5}

Multiply by the conjugate

\frac{5}{5}

= - \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= - \frac{5}{5}

= - \frac{5}{5}

cos (x) = - \frac{5}{5}

cos (x) = - \frac{5}{5}

cos (x) = - \frac{5}{5}

Apply trig inverse properties

cos (x) = - \frac{5}{5}

General solutions for

cos (x) = - \frac{5}{5}

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

x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

tan (x) - 5 sin (x) = 0 : x = 2 πn, x = π + 2 πn, x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

tan (x) - 5 sin (x) = 0

Express with sin, cos

tan (x) - sin (x) 5

Use the basic trigonometric identity:

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

= \frac{sin ( x )}{cos ( x )} - sin (x) 5

Simplify

\frac{sin ( x )}{cos ( x )} - sin (x) 5 : \frac{sin ( x ) - 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x )}{cos ( x )} - sin (x) 5

Convert element to fraction:

5 sin (x) = \frac{sin ( x ) 5 cos ( x )}{cos ( x )}

= \frac{sin ( x )}{cos ( x )} - \frac{sin ( x ) 5 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 ) - sin ( x ) 5 cos ( x )}{cos ( x )}

= \frac{sin ( x ) - 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x ) - cos ( x ) sin ( x ) 5}{cos ( x )} = 0

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

sin (x) - cos (x) sin (x) 5 = 0

Factor

sin (x) - cos (x) sin (x) 5 : sin (x) (1 - 5 cos (x))

sin (x) - cos (x) sin (x) 5

Factor out common term

sin (x)

= sin (x) (1 - cos (x) 5)

sin (x) (1 - 5 cos (x)) = 0

Solving each part separately

sin (x) = 0 or 1 - 5 cos (x) = 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

1 - 5 cos (x) = 0 : x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

1 - 5 cos (x) = 0

Move

1

to the right side

1 - 5 cos (x) = 0

Subtract

1

from both sides

1 - 5 cos (x) - 1 = 0 - 1

Simplify

- 5 cos (x) = - 1

- 5 cos (x) = - 1

Divide both sides by

- 5

- 5 cos (x) = - 1

Divide both sides by

- 5

\frac{- 5 cos ( x )}{- 5} = \frac{- 1}{- 5}

Simplify

\frac{- 5 cos ( x )}{- 5} = \frac{- 1}{- 5}

Simplify

\frac{- 5 cos ( x )}{- 5} : cos (x)

\frac{- 5 cos ( x )}{- 5}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{5 cos ( x )}{5}

Cancel the common factor:

5

= cos (x)

Simplify

\frac{- 1}{- 5} : \frac{5}{5}

\frac{- 1}{- 5}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{1}{5}

Rationalize

\frac{1}{5} : \frac{5}{5}

\frac{1}{5}

Multiply by the conjugate

\frac{5}{5}

= \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= \frac{5}{5}

= \frac{5}{5}

cos (x) = \frac{5}{5}

cos (x) = \frac{5}{5}

cos (x) = \frac{5}{5}

Apply trig inverse properties

cos (x) = \frac{5}{5}

General solutions for

cos (x) = \frac{5}{5}

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

x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn, x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 2.03444 \dots + 2 πn, x = - 2.03444 \dots + 2 πn, x = 1.10714 \dots + 2 πn, x = 2 π - 1.10714 \dots + 2 πn

Graph

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