What is the general solution for cos(x)=sqrt(2)cos(45+x) ?

The general solution for cos(x)=sqrt(2)cos(45+x) is x=360n,x=180+360n

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cos(x)=sqrt(2)cos(45+x)

Solution

cos (x) = 2 cos (4 5^{\circ} + x)

Solution

x = 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Radians

x = 0 + 2 πn, x = π + 2 πn

Solution steps

cos (x) = 2 cos (4 5^{\circ} + x)

Rewrite using trig identities

cos (x) = 2 cos (4 5^{\circ} + x)

Rewrite using trig identities

cos (4 5^{\circ} + x)

Use the Angle Sum identity:

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

= cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x)

Simplify

cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x) : \frac{2 cos ( x ) - 2 sin ( x )}{2}

cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x)

cos (4 5^{\circ}) cos (x) = \frac{2 cos ( x )}{2}

cos (4 5^{\circ}) cos (x)

Simplify

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

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

Multiply fractions:

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

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

sin (4 5^{\circ}) sin (x) = \frac{2 sin ( x )}{2}

sin (4 5^{\circ}) sin (x)

Simplify

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

Multiply fractions:

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

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

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

Apply rule

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

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

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

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

Simplify

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

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

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

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

Factor out common term

2

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

Cancel

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

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

Apply radical rule:

2 = 2^{\frac{1}{2}}

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

2^{\frac{1}{2}} = 2

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= cos (x) - sin (x)

cos (x) = cos (x) - sin (x)

cos (x) = cos (x) - sin (x)

Subtract

cos (x) - sin (x)

from both sides

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

x = 0 + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solve

x = 0 + 36 0^{\circ} n : x = 36 0^{\circ} n

x = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

x = 36 0^{\circ} n

x = 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

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tan^{2} (θ) - 16 = 0

Frequently Asked Questions (FAQ)

What is the general solution for cos(x)=sqrt(2)cos(45+x) ?
The general solution for cos(x)=sqrt(2)cos(45+x) is x=360n,x=180+360n