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Popular Trigonometry >

2cos^2(a)tan(a)=tan(a)

Solution

2 cos^{2} (a) tan (a) = tan (a)

Solution

a = πn, a = \frac{3 π}{4} + 2 πn, a = \frac{5 π}{4} + 2 πn, a = \frac{π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

Degrees

a = 0^{\circ} + 18 0^{\circ} n, a = 13 5^{\circ} + 36 0^{\circ} n, a = 22 5^{\circ} + 36 0^{\circ} n, a = 4 5^{\circ} + 36 0^{\circ} n, a = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

2 cos^{2} (a) tan (a) = tan (a)

Subtract

tan (a)

from both sides

2 cos^{2} (a) tan (a) - tan (a) = 0

Factor

2 cos^{2} (a) tan (a) - tan (a) : tan (a) (2 cos (a) + 1) (2 cos (a) - 1)

2 cos^{2} (a) tan (a) - tan (a)

Factor out common term

tan (a)

= tan (a) (2 cos^{2} (a) - 1)

Factor

2 cos^{2} (a) - 1 : (2 cos (a) + 1) (2 cos (a) - 1)

2 cos^{2} (a) - 1

Rewrite

2 cos^{2} (a) - 1

(2 cos (a))^{2} - 1^{2}

2 cos^{2} (a) - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} cos^{2} (a) - 1

Rewrite

1

1^{2}

= (2)^{2} cos^{2} (a) - 1^{2}

Apply exponent rule:

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

(2)^{2} cos^{2} (a) = (2 cos (a))^{2}

= (2 cos (a))^{2} - 1^{2}

= (2 cos (a))^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

(2 cos (a))^{2} - 1^{2} = (2 cos (a) + 1) (2 cos (a) - 1)

= (2 cos (a) + 1) (2 cos (a) - 1)

= tan (a) (2 cos (a) + 1) (2 cos (a) - 1)

tan (a) (2 cos (a) + 1) (2 cos (a) - 1) = 0

Solving each part separately

tan (a) = 0 or 2 cos (a) + 1 = 0 or 2 cos (a) - 1 = 0

tan (a) = 0 : a = πn

tan (a) = 0

General solutions for

tan (a) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

a = 0 + πn

a = 0 + πn

Solve

a = 0 + πn : a = πn

a = 0 + πn

0 + πn = πn

a = πn

a = πn

2 cos (a) + 1 = 0 : a = \frac{3 π}{4} + 2 πn, a = \frac{5 π}{4} + 2 πn

2 cos (a) + 1 = 0

Move

1

to the right side

2 cos (a) + 1 = 0

Subtract

1

from both sides

2 cos (a) + 1 - 1 = 0 - 1

Simplify

2 cos (a) = - 1

2 cos (a) = - 1

Divide both sides by

2

2 cos (a) = - 1

Divide both sides by

2

\frac{2 cos ( a )}{2} = \frac{- 1}{2}

Simplify

\frac{2 cos ( a )}{2} = \frac{- 1}{2}

Simplify

\frac{2 cos ( a )}{2} : cos (a)

\frac{2 cos ( a )}{2}

Cancel the common factor:

2

= cos (a)

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

cos (a) = - \frac{2}{2}

cos (a) = - \frac{2}{2}

cos (a) = - \frac{2}{2}

General solutions for

cos (a) = - \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = \frac{3 π}{4} + 2 πn, a = \frac{5 π}{4} + 2 πn

a = \frac{3 π}{4} + 2 πn, a = \frac{5 π}{4} + 2 πn

2 cos (a) - 1 = 0 : a = \frac{π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

2 cos (a) - 1 = 0

Move

1

to the right side

2 cos (a) - 1 = 0

Add

1

to both sides

2 cos (a) - 1 + 1 = 0 + 1

Simplify

2 cos (a) = 1

2 cos (a) = 1

Divide both sides by

2

2 cos (a) = 1

Divide both sides by

2

\frac{2 cos ( a )}{2} = \frac{1}{2}

Simplify

\frac{2 cos ( a )}{2} = \frac{1}{2}

Simplify

\frac{2 cos ( a )}{2} : cos (a)

\frac{2 cos ( a )}{2}

Cancel the common factor:

2

= cos (a)

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

cos (a) = \frac{2}{2}

cos (a) = \frac{2}{2}

cos (a) = \frac{2}{2}

General solutions for

cos (a) = \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = \frac{π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

a = \frac{π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

Combine all the solutions

a = πn, a = \frac{3 π}{4} + 2 πn, a = \frac{5 π}{4} + 2 πn, a = \frac{π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

Graph

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