What is the general solution for cos(7x)cos(2x)+sin(7x)sin(2x)= 1/2 ?

The general solution for cos(7x)cos(2x)+sin(7x)sin(2x)= 1/2 is x= pi/(15)+(2pin)/5 ,x= pi/3+(2pin)/5

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cos(7x)cos(2x)+sin(7x)sin(2x)= 1/2

Solution

cos (7 x) cos (2 x) + sin (7 x) sin (2 x) = \frac{1}{2}

Solution

x = \frac{π}{15} + \frac{2 πn}{5}, x = \frac{π}{3} + \frac{2 πn}{5}

Degrees

x = 1 2^{\circ} + 7 2^{\circ} n, x = 6 0^{\circ} + 7 2^{\circ} n

Solution steps

cos (7 x) cos (2 x) + sin (7 x) sin (2 x) = \frac{1}{2}

Rewrite using trig identities

cos (7 x) cos (2 x) + sin (7 x) sin (2 x)

Use the Angle Difference identity:

cos (s) cos (t) + sin (s) sin (t) = cos (s - t)

= cos (7 x - 2 x)

cos (7 x - 2 x) = \frac{1}{2}

General solutions for

cos (7 x - 2 x) = \frac{1}{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}

7 x - 2 x = \frac{π}{3} + 2 πn, 7 x - 2 x = \frac{5 π}{3} + 2 πn

7 x - 2 x = \frac{π}{3} + 2 πn, 7 x - 2 x = \frac{5 π}{3} + 2 πn

Solve

7 x - 2 x = \frac{π}{3} + 2 πn : x = \frac{π}{15} + \frac{2 πn}{5}

7 x - 2 x = \frac{π}{3} + 2 πn

Add similar elements:

7 x - 2 x = 5 x

5 x = \frac{π}{3} + 2 πn

Divide both sides by

5

5 x = \frac{π}{3} + 2 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{π}{3}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{π}{3}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{π}{3}}{5} + \frac{2 πn}{5} : \frac{π}{15} + \frac{2 πn}{5}

\frac{\frac{π}{3}}{5} + \frac{2 πn}{5}

\frac{\frac{π}{3}}{5} = \frac{π}{15}

\frac{\frac{π}{3}}{5}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{π}{15}

= \frac{π}{15} + \frac{2 πn}{5}

x = \frac{π}{15} + \frac{2 πn}{5}

x = \frac{π}{15} + \frac{2 πn}{5}

x = \frac{π}{15} + \frac{2 πn}{5}

Solve

7 x - 2 x = \frac{5 π}{3} + 2 πn : x = \frac{π}{3} + \frac{2 πn}{5}

7 x - 2 x = \frac{5 π}{3} + 2 πn

Add similar elements:

7 x - 2 x = 5 x

5 x = \frac{5 π}{3} + 2 πn

Divide both sides by

5

5 x = \frac{5 π}{3} + 2 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5} : \frac{π}{3} + \frac{2 πn}{5}

\frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5}

\frac{\frac{5 π}{3}}{5} = \frac{π}{3}

\frac{\frac{5 π}{3}}{5}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{5 π}{15}

Cancel the common factor:

5

= \frac{π}{3}

= \frac{π}{3} + \frac{2 πn}{5}

x = \frac{π}{3} + \frac{2 πn}{5}

x = \frac{π}{3} + \frac{2 πn}{5}

x = \frac{π}{3} + \frac{2 πn}{5}

x = \frac{π}{15} + \frac{2 πn}{5}, x = \frac{π}{3} + \frac{2 πn}{5}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(7x)cos(2x)+sin(7x)sin(2x)= 1/2 ?
The general solution for cos(7x)cos(2x)+sin(7x)sin(2x)= 1/2 is x= pi/(15)+(2pin)/5 ,x= pi/3+(2pin)/5