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Popular Trigonometry >

cot^2(a)=cos^2(a)+cos(a)cos(a)

Solution

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

Solution

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

Degrees

a = 9 0^{\circ} + 36 0^{\circ} n, a = 27 0^{\circ} + 36 0^{\circ} n, a = 22 5^{\circ} + 36 0^{\circ} n, a = 31 5^{\circ} + 36 0^{\circ} n, a = 4 5^{\circ} + 36 0^{\circ} n, a = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

cot (a) + 2 cos (a) = 0 or cot (a) - 2 cos (a) = 0

cot (a) + 2 cos (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn, a = \frac{5 π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

cot (a) + 2 cos (a) = 0

Express with sin, cos

cot (a) + cos (a) 2

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

\frac{cos ( a ) + cos ( a ) sin ( a ) 2}{sin ( a )} = 0

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

cos (a) + cos (a) sin (a) 2 = 0

Factor

cos (a) + cos (a) sin (a) 2 : cos (a) (1 + 2 sin (a))

cos (a) + cos (a) sin (a) 2

Factor out common term

cos (a)

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

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

Solving each part separately

cos (a) = 0 or 1 + 2 sin (a) = 0

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

cos (a) = 0

General solutions for

cos (a) = 0

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}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

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

1 + 2 sin (a) = 0 : a = \frac{5 π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

1 + 2 sin (a) = 0

Move

1

to the right side

1 + 2 sin (a) = 0

Subtract

1

from both sides

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

Simplify

2 sin (a) = - 1

2 sin (a) = - 1

Divide both sides by

2

2 sin (a) = - 1

Divide both sides by

2

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

Simplify

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

Simplify

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

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

Cancel the common factor:

2

= sin (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}

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

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

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

General solutions for

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

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}

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

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

Combine all the solutions

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

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

cot (a) - 2 cos (a) = 0

Express with sin, cos

cot (a) - cos (a) 2

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

\frac{cos ( a ) - cos ( a ) sin ( a ) 2}{sin ( a )} = 0

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

cos (a) - cos (a) sin (a) 2 = 0

Factor

cos (a) - cos (a) sin (a) 2 : cos (a) (1 - 2 sin (a))

cos (a) - cos (a) sin (a) 2

Factor out common term

cos (a)

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

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

Solving each part separately

cos (a) = 0 or 1 - 2 sin (a) = 0

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

cos (a) = 0

General solutions for

cos (a) = 0

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}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

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

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

1 - 2 sin (a) = 0

Move

1

to the right side

1 - 2 sin (a) = 0

Subtract

1

from both sides

1 - 2 sin (a) - 1 = 0 - 1

Simplify

- 2 sin (a) = - 1

- 2 sin (a) = - 1

Divide both sides by

- 2

- 2 sin (a) = - 1

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 sin ( a )}{- 2} : sin (a)

\frac{- 2 sin ( a )}{- 2}

Apply the fraction rule:

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

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

Cancel the common factor:

2

= sin (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}

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

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

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

General solutions for

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

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}

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

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

Combine all the solutions

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

Combine all the solutions

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

Graph

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