What is the general solution for cos(A-25)+cos(A+25)=cos(25) ?

The general solution for cos(A-25)+cos(A+25)=cos(25) is A=60+360n,A=300+360n

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cos(A-25)+cos(A+25)=cos(25)

Solution

cos (A - 2 5^{\circ}) + cos (A + 2 5^{\circ}) = cos (2 5^{\circ})

Solution

A = 6 0^{\circ} + 36 0^{\circ} n, A = 30 0^{\circ} + 36 0^{\circ} n

Radians

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

Solution steps

cos (A - 2 5^{\circ}) + cos (A + 2 5^{\circ}) = cos (2 5^{\circ})

Rewrite using trig identities

cos (A - 2 5^{\circ}) + cos (A + 2 5^{\circ})

Use the Sum to Product identity:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 cos (\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2}) cos (\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2})

Simplify

2 cos (\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2}) cos (\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2}) : 2 cos (2 5^{\circ}) cos (A)

2 cos (\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2}) cos (\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2})

\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2} = A

\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2}

A - 2 5^{\circ} + A + 2 5^{\circ} = 2 A

A - 2 5^{\circ} + A + 2 5^{\circ}

Group like terms

= A + A - 2 5^{\circ} + 2 5^{\circ}

Add similar elements:

A + A = 2 A

= 2 A - 2 5^{\circ} + 2 5^{\circ}

Add similar elements:

- 2 5^{\circ} + 2 5^{\circ} = 0

= 2 A

= \frac{2 A}{2}

Divide the numbers:

\frac{2}{2} = 1

= A

= 2 cos (A) cos (\frac{A - ( A + 2 5 ^{\circ} ) - 2 5 ^{\circ}}{2})

\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2} = - 2 5^{\circ}

\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2}

Join

A - 2 5^{\circ} - (A + 2 5^{\circ}) : - 5 0^{\circ}

A - 2 5^{\circ} - (A + 2 5^{\circ})

Convert element to fraction:

A = \frac{A 36}{36}, (A + 2 5^{\circ}) = \frac{( A + 2 5 ^{\circ} ) 36}{36}

= \frac{A \cdot 36}{36} - 2 5^{\circ} - \frac{( A + 2 5 ^{\circ} ) \cdot 36}{36}

Since the denominators are equal, combine the fractions:

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

= \frac{A \cdot 36 - 90 0 ^{\circ} - ( A + 2 5 ^{\circ} ) \cdot 36}{36}

Expand

A \cdot 36 - 90 0^{\circ} - (A + 2 5^{\circ}) \cdot 36 : - 180 0^{\circ}

A \cdot 36 - 90 0^{\circ} - (A + 2 5^{\circ}) \cdot 36

= 36 A - 90 0^{\circ} - 36 (A + 2 5^{\circ})

Expand

- 36 (A + 2 5^{\circ}) : - 36 A - 90 0^{\circ}

- 36 (A + 2 5^{\circ})

Apply the distributive law:

a (b + c) = ab + a c

a = - 36, b = A, c = 2 5^{\circ}

= - 36 A + (- 36) 2 5^{\circ}

Apply minus-plus rules

+ (- a) = - a

= - 36 A - 36 \cdot 2 5^{\circ}

36 \cdot 2 5^{\circ} = 90 0^{\circ}

36 \cdot 2 5^{\circ}

Multiply fractions:

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

= 90 0^{\circ}

Cancel the common factor:

36

= 90 0^{\circ}

= - 36 A - 90 0^{\circ}

= A \cdot 36 - 90 0^{\circ} - 36 A - 90 0^{\circ}

Simplify

A \cdot 36 - 90 0^{\circ} - 36 A - 90 0^{\circ} : - 180 0^{\circ}

A \cdot 36 - 90 0^{\circ} - 36 A - 90 0^{\circ}

Group like terms

= 36 A - 36 A - 90 0^{\circ} - 90 0^{\circ}

Add similar elements:

36 A - 36 A = 0

= - 90 0^{\circ} - 90 0^{\circ}

Add similar elements:

- 90 0^{\circ} - 90 0^{\circ} = - 180 0^{\circ}

= - 180 0^{\circ}

= - 180 0^{\circ}

= \frac{- 180 0 ^{\circ}}{36}

Apply the fraction rule:

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

= - 5 0^{\circ}

Cancel the common factor:

2

= - 5 0^{\circ}

= \frac{- 5 0 ^{\circ}}{2}

Apply the fraction rule:

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

= - \frac{5 0 ^{\circ}}{2}

Apply the fraction rule:

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

\frac{5 0 ^{\circ}}{2} = \frac{90 0 ^{\circ}}{18 \cdot 2}

= - \frac{90 0 ^{\circ}}{18 \cdot 2}

Multiply the numbers:

18 \cdot 2 = 36

= - 2 5^{\circ}

= 2 cos (- 2 5^{\circ}) cos (A)

Simplify

cos (- 2 5^{\circ}) : cos (2 5^{\circ})

cos (- 2 5^{\circ})

Use the following property:

cos (- x) = cos (x)

cos (- 2 5^{\circ}) = cos (2 5^{\circ})

= cos (2 5^{\circ})

= 2 cos (2 5^{\circ}) cos (A)

= 2 cos (2 5^{\circ}) cos (A)

2 cos (2 5^{\circ}) cos (A) = cos (2 5^{\circ})

Divide both sides by

2 cos (2 5^{\circ})

2 cos (2 5^{\circ}) cos (A) = cos (2 5^{\circ})

Divide both sides by

2 cos (2 5^{\circ})

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

Simplify

cos (A) = \frac{1}{2}

cos (A) = \frac{1}{2}

General solutions for

cos (A) = \frac{1}{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}

A = 6 0^{\circ} + 36 0^{\circ} n, A = 30 0^{\circ} + 36 0^{\circ} n

A = 6 0^{\circ} + 36 0^{\circ} n, A = 30 0^{\circ} + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for cos(A-25)+cos(A+25)=cos(25) ?
The general solution for cos(A-25)+cos(A+25)=cos(25) is A=60+360n,A=300+360n