What is the general solution for sin(x)cos(x)=-1/4 ?

The general solution for sin(x)cos(x)=-1/4 is x=(7pi)/(12)+pin,x=(11pi)/(12)+pin

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sin(x)cos(x)=-1/4

Solution

sin (x) cos (x) = - \frac{1}{4}

Solution

x = \frac{7 π}{12} + πn, x = \frac{11 π}{12} + πn

Degrees

x = 10 5^{\circ} + 18 0^{\circ} n, x = 16 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (x) cos (x) = - \frac{1}{4}

Rewrite using trig identities

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}

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

\frac{sin ( 2 x )}{2} = - \frac{1}{4}

Multiply both sides by

2

\frac{sin ( 2 x )}{2} = - \frac{1}{4}

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} = 2 (- \frac{1}{4})

Simplify

\frac{2 sin ( 2 x )}{2} = 2 (- \frac{1}{4})

Simplify

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

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

Divide the numbers:

\frac{2}{2} = 1

= sin (2 x)

Simplify

2 (- \frac{1}{4}) : - \frac{1}{2}

2 (- \frac{1}{4})

Remove parentheses:

(- a) = - a

= - 2 \cdot \frac{1}{4}

Multiply fractions:

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

= - \frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

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

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

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

General solutions for

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

2 x = \frac{7 π}{6} + 2 πn, 2 x = \frac{11 π}{6} + 2 πn

2 x = \frac{7 π}{6} + 2 πn, 2 x = \frac{11 π}{6} + 2 πn

Solve

2 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{12} + πn

2 x = \frac{7 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{7 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2} : \frac{7 π}{12} + πn

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{6}}{2} = \frac{7 π}{12}

\frac{\frac{7 π}{6}}{2}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{7 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

Solve

2 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{12} + πn

2 x = \frac{11 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{11 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2} : \frac{11 π}{12} + πn

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{11 π}{6}}{2} = \frac{11 π}{12}

\frac{\frac{11 π}{6}}{2}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{11 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{7 π}{12} + πn, x = \frac{11 π}{12} + πn

Popular Examples

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sec^{2} (θ) - 9 = 0

2cos^2(θ)=2-sin(θ)

2 cos^{2} (θ) = 2 - sin (θ)

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cos (x + 1) = 0

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sec^{2} (x) + tan^{2} (x) = 7

4cos(x)=1,0<= x<= 2pi

4 cos (x) = 1, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)cos(x)=-1/4 ?
The general solution for sin(x)cos(x)=-1/4 is x=(7pi)/(12)+pin,x=(11pi)/(12)+pin