What is the general solution for cos(5a+40)= 3/5 ?

The general solution for cos(5a+40)= 3/5 is a=(360n)/5-8+(0.92729…)/5 ,a=72-8+(360n)/5-(0.92729…)/5

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cos(5a+40)= 3/5

Solution

cos (5 a + 4 0^{\circ}) = \frac{3}{5}

Solution

a = \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{0.92729 \dots}{5}, a = 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{0.92729 \dots}{5}

Radians

a = - \frac{2 π}{45} + \frac{0.92729 \dots}{5} + \frac{2 π}{5} n, a = \frac{2 π}{5} - \frac{2 π}{45} - \frac{0.92729 \dots}{5} + \frac{2 π}{5} n

Solution steps

cos (5 a + 4 0^{\circ}) = \frac{3}{5}

Apply trig inverse properties

cos (5 a + 4 0^{\circ}) = \frac{3}{5}

General solutions for

cos (5 a + 4 0^{\circ}) = \frac{3}{5}

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

5 a + 4 0^{\circ} = arccos (\frac{3}{5}) + 36 0^{\circ} n, 5 a + 4 0^{\circ} = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n

5 a + 4 0^{\circ} = arccos (\frac{3}{5}) + 36 0^{\circ} n, 5 a + 4 0^{\circ} = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n

Solve

5 a + 4 0^{\circ} = arccos (\frac{3}{5}) + 36 0^{\circ} n : a = \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{arccos ( \frac{3}{5} )}{5}

5 a + 4 0^{\circ} = arccos (\frac{3}{5}) + 36 0^{\circ} n

Move

4 0^{\circ}

to the right side

5 a + 4 0^{\circ} = arccos (\frac{3}{5}) + 36 0^{\circ} n

Subtract

4 0^{\circ}

from both sides

5 a + 4 0^{\circ} - 4 0^{\circ} = arccos (\frac{3}{5}) + 36 0^{\circ} n - 4 0^{\circ}

Simplify

5 a = arccos (\frac{3}{5}) + 36 0^{\circ} n - 4 0^{\circ}

5 a = arccos (\frac{3}{5}) + 36 0^{\circ} n - 4 0^{\circ}

Divide both sides by

5

5 a = arccos (\frac{3}{5}) + 36 0^{\circ} n - 4 0^{\circ}

Divide both sides by

5

\frac{5 a}{5} = \frac{arccos ( \frac{3}{5} )}{5} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5}

Simplify

\frac{5 a}{5} = \frac{arccos ( \frac{3}{5} )}{5} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5}

Simplify

\frac{5 a}{5} : a

\frac{5 a}{5}

Divide the numbers:

\frac{5}{5} = 1

= a

Simplify

\frac{arccos ( \frac{3}{5} )}{5} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5} : \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{arccos ( \frac{3}{5} )}{5}

\frac{arccos ( \frac{3}{5} )}{5} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5}

Group like terms

= \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5} + \frac{arccos ( \frac{3}{5} )}{5}

\frac{4 0 ^{\circ}}{5} = 8^{\circ}

\frac{4 0 ^{\circ}}{5}

Apply the fraction rule:

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

= \frac{36 0 ^{\circ}}{9 \cdot 5}

Multiply the numbers:

9 \cdot 5 = 45

= 8^{\circ}

= \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{arccos ( \frac{3}{5} )}{5}

a = \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{arccos ( \frac{3}{5} )}{5}

a = \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{arccos ( \frac{3}{5} )}{5}

a = \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{arccos ( \frac{3}{5} )}{5}

Solve

5 a + 4 0^{\circ} = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n : a = 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{arccos ( \frac{3}{5} )}{5}

5 a + 4 0^{\circ} = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n

Move

4 0^{\circ}

to the right side

5 a + 4 0^{\circ} = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n

Subtract

4 0^{\circ}

from both sides

5 a + 4 0^{\circ} - 4 0^{\circ} = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n - 4 0^{\circ}

Simplify

5 a = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n - 4 0^{\circ}

5 a = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n - 4 0^{\circ}

Divide both sides by

5

5 a = 36 0^{\circ} - arccos (\frac{3}{5}) + 36 0^{\circ} n - 4 0^{\circ}

Divide both sides by

5

\frac{5 a}{5} = 7 2^{\circ} - \frac{arccos ( \frac{3}{5} )}{5} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5}

Simplify

\frac{5 a}{5} = 7 2^{\circ} - \frac{arccos ( \frac{3}{5} )}{5} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5}

Simplify

\frac{5 a}{5} : a

\frac{5 a}{5}

Divide the numbers:

\frac{5}{5} = 1

= a

Simplify

7 2^{\circ} - \frac{arccos ( \frac{3}{5} )}{5} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5} : 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{arccos ( \frac{3}{5} )}{5}

7 2^{\circ} - \frac{arccos ( \frac{3}{5} )}{5} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5}

Group like terms

= 7 2^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{4 0 ^{\circ}}{5} - \frac{arccos ( \frac{3}{5} )}{5}

\frac{4 0 ^{\circ}}{5} = 8^{\circ}

\frac{4 0 ^{\circ}}{5}

Apply the fraction rule:

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

= \frac{36 0 ^{\circ}}{9 \cdot 5}

Multiply the numbers:

9 \cdot 5 = 45

= 8^{\circ}

= 7 2^{\circ} + \frac{36 0 ^{\circ} n}{5} - 8^{\circ} - \frac{arccos ( \frac{3}{5} )}{5}

Group like terms

= 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{arccos ( \frac{3}{5} )}{5}

a = 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{arccos ( \frac{3}{5} )}{5}

a = 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{arccos ( \frac{3}{5} )}{5}

a = 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{arccos ( \frac{3}{5} )}{5}

a = \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{arccos ( \frac{3}{5} )}{5}, a = 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{arccos ( \frac{3}{5} )}{5}

Show solutions in decimal form

a = \frac{36 0 ^{\circ} n}{5} - 8^{\circ} + \frac{0.92729 \dots}{5}, a = 7 2^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{5} - \frac{0.92729 \dots}{5}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(5a+40)= 3/5 ?
The general solution for cos(5a+40)= 3/5 is a=(360n)/5-8+(0.92729…)/5 ,a=72-8+(360n)/5-(0.92729…)/5