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Popular Trigonometry >

1+cos^2(a)=3sin(x)cos(x)

Solution

1 + cos^{2} (a) = 3 sin (x) cos (x)

Solution

x = \frac{arcsin ( - \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn

Solution steps

1 + cos^{2} (a) = 3 sin (x) cos (x)

Subtract

3 sin (x) cos (x)

from both sides

1 + cos^{2} (a) - 3 sin (x) cos (x) = 0

Rewrite using trig identities

1 + cos^{2} (a) - 3 sin (x) cos (x)

Use the Double Angle identity:

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

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

= 1 + cos^{2} (a) - 3 \cdot \frac{sin ( 2 x )}{2}

1 + cos^{2} (a) - 3 \cdot \frac{sin ( 2 x )}{2} = 0

Move

cos^{2} (a)

to the right side

1 + cos^{2} (a) - 3 \cdot \frac{sin ( 2 x )}{2} = 0

Subtract

cos^{2} (a)

from both sides

1 + cos^{2} (a) - 3 \cdot \frac{sin ( 2 x )}{2} - cos^{2} (a) = 0 - cos^{2} (a)

Simplify

1 - 3 \cdot \frac{sin ( 2 x )}{2} = - cos^{2} (a)

1 - 3 \cdot \frac{sin ( 2 x )}{2} = - cos^{2} (a)

Move

1

to the right side

1 - 3 \cdot \frac{sin ( 2 x )}{2} = - cos^{2} (a)

Subtract

1

from both sides

1 - 3 \cdot \frac{sin ( 2 x )}{2} - 1 = - cos^{2} (a) - 1

Simplify

- 3 \cdot \frac{sin ( 2 x )}{2} = - cos^{2} (a) - 1

- 3 \cdot \frac{sin ( 2 x )}{2} = - cos^{2} (a) - 1

Refine

- 3 \cdot \frac{sin ( 2 x )}{2} : - \frac{3 sin ( 2 x )}{2}

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

Multiply fractions:

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

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

- \frac{3 sin ( 2 x )}{2} = - cos^{2} (a) - 1

Multiply both sides by

2

- \frac{3 sin ( 2 x )}{2} = - cos^{2} (a) - 1

Multiply both sides by

2

- \frac{3 sin ( 2 x )}{2} \cdot 2 = - cos^{2} (a) \cdot 2 - 1 \cdot 2

Simplify

- \frac{3 sin ( 2 x )}{2} \cdot 2 = - cos^{2} (a) \cdot 2 - 1 \cdot 2

Simplify

- \frac{3 sin ( 2 x )}{2} \cdot 2 : - 3 sin (2 x)

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - 3 sin (2 x)

Simplify

cos^{2} (a) \cdot 2 : 2 cos^{2} (a)

cos^{2} (a) \cdot 2

Apply the commutative law:

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

2 cos^{2} (a)

Simplify

- 1 \cdot 2 : - 2

- 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= - 2

- 3 sin (2 x) = - 2 cos^{2} (a) - 2

- 3 sin (2 x) = - 2 cos^{2} (a) - 2

- 3 sin (2 x) = - 2 cos^{2} (a) - 2

Divide both sides by

- 3

- 3 sin (2 x) = - 2 cos^{2} (a) - 2

Divide both sides by

- 3

\frac{- 3 sin ( 2 x )}{- 3} = - \frac{2 cos ^{2} ( a )}{- 3} - \frac{2}{- 3}

Simplify

\frac{- 3 sin ( 2 x )}{- 3} = - \frac{2 cos ^{2} ( a )}{- 3} - \frac{2}{- 3}

Simplify

\frac{- 3 sin ( 2 x )}{- 3} : sin (2 x)

\frac{- 3 sin ( 2 x )}{- 3}

Apply the fraction rule:

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

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

Divide the numbers:

\frac{3}{3} = 1

= sin (2 x)

Simplify

- \frac{2 cos ^{2} ( a )}{- 3} - \frac{2}{- 3} : - \frac{- 2 cos ^{2} ( a ) - 2}{3}

- \frac{2 cos ^{2} ( a )}{- 3} - \frac{2}{- 3}

Apply rule

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

= \frac{- 2 cos ^{2} ( a ) - 2}{- 3}

Apply the fraction rule:

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

= - \frac{- 2 cos ^{2} ( a ) - 2}{3}

sin (2 x) = - \frac{- 2 cos ^{2} ( a ) - 2}{3}

sin (2 x) = - \frac{- 2 cos ^{2} ( a ) - 2}{3}

sin (2 x) = - \frac{- 2 cos ^{2} ( a ) - 2}{3}

Apply trig inverse properties

sin (2 x) = - \frac{- 2 cos ^{2} ( a ) - 2}{3}

General solutions for

sin (2 x) = - \frac{- 2 cos ^{2} ( a ) - 2}{3}

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

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

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

Solve

2 x = arcsin (- \frac{- 2 cos ^{2} ( a ) - 2}{3}) + 2 πn : x = \frac{arcsin ( - \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn

2 x = arcsin (- \frac{- 2 cos ^{2} ( a ) - 2}{3}) + 2 πn

Divide both sides by

2

2 x = arcsin (- \frac{- 2 cos ^{2} ( a ) - 2}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsin ( - \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arcsin ( - \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn

x = \frac{arcsin ( - \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn

Solve

2 x = π + arcsin (\frac{- 2 cos ^{2} ( a ) - 2}{3}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn

2 x = π + arcsin (\frac{- 2 cos ^{2} ( a ) - 2}{3}) + 2 πn

Divide both sides by

2

2 x = π + arcsin (\frac{- 2 cos ^{2} ( a ) - 2}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + \frac{arcsin ( \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn

x = \frac{arcsin ( - \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{- 2 c o s ^{2} ( a ) - 2}{3} )}{2} + πn

Graph

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