What is the general solution for 3cos(45)+4cos(y)=3 ?

The general solution for 3cos(45)+4cos(y)=3 is y=1.34932…+360n,y=360-1.34932…+360n

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3cos(45)+4cos(y)=3

Solution

3 cos (4 5^{\circ}) + 4 cos (y) = 3

Solution

y = 1.34932 \dots + 36 0^{\circ} n, y = 36 0^{\circ} - 1.34932 \dots + 36 0^{\circ} n

Radians

y = 1.34932 \dots + 2 πn, y = 2 π - 1.34932 \dots + 2 πn

Solution steps

3 cos (4 5^{\circ}) + 4 cos (y) = 3

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

cos (4 5^{\circ})

Use the following trivial identity:

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

cos (4 5^{\circ})

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}

3 \cdot \frac{2}{2} + 4 cos (y) = 3

Move

3 \frac{2}{2}

to the right side

3 \cdot \frac{2}{2} + 4 cos (y) = 3

Subtract

3 \frac{2}{2}

from both sides

3 \cdot \frac{2}{2} + 4 cos (y) - 3 \cdot \frac{2}{2} = 3 - 3 \cdot \frac{2}{2}

Simplify

3 \cdot \frac{2}{2} + 4 cos (y) - 3 \cdot \frac{2}{2} = 3 - 3 \cdot \frac{2}{2}

Simplify

3 \cdot \frac{2}{2} + 4 cos (y) - 3 \cdot \frac{2}{2} : 4 cos (y)

3 \cdot \frac{2}{2} + 4 cos (y) - 3 \cdot \frac{2}{2}

Add similar elements:

3 \frac{2}{2} - 3 \frac{2}{2} = 0

= 4 cos (y)

Simplify

3 - 3 \cdot \frac{2}{2} : 3 - \frac{3 2}{2}

3 - 3 \cdot \frac{2}{2}

Multiply

3 \cdot \frac{2}{2} : \frac{3 2}{2}

3 \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 3}{2}

= 3 - \frac{3 2}{2}

4 cos (y) = 3 - \frac{3 2}{2}

4 cos (y) = 3 - \frac{3 2}{2}

4 cos (y) = 3 - \frac{3 2}{2}

Divide both sides by

4

4 cos (y) = 3 - \frac{3 2}{2}

Divide both sides by

4

\frac{4 cos ( y )}{4} = \frac{3}{4} - \frac{\frac{3 2}{2}}{4}

Simplify

\frac{4 cos ( y )}{4} = \frac{3}{4} - \frac{\frac{3 2}{2}}{4}

Simplify

\frac{4 cos ( y )}{4} : cos (y)

\frac{4 cos ( y )}{4}

Divide the numbers:

\frac{4}{4} = 1

= cos (y)

Simplify

\frac{3}{4} - \frac{\frac{3 2}{2}}{4} : \frac{6 - 3 2}{8}

\frac{3}{4} - \frac{\frac{3 2}{2}}{4}

Apply rule

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

= \frac{3 - \frac{3 2}{2}}{4}

Join

3 - \frac{3 2}{2} : \frac{6 - 3 2}{2}

3 - \frac{3 2}{2}

Convert element to fraction:

3 = \frac{3 \cdot 2}{2}

= \frac{3 \cdot 2}{2} - \frac{3 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{3 \cdot 2 - 3 2}{2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 - 3 2}{2}

= \frac{\frac{6 - 3 2}{2}}{4}

Apply the fraction rule:

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

= \frac{6 - 3 2}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{6 - 3 2}{8}

cos (y) = \frac{6 - 3 2}{8}

cos (y) = \frac{6 - 3 2}{8}

cos (y) = \frac{6 - 3 2}{8}

Apply trig inverse properties

cos (y) = \frac{6 - 3 2}{8}

General solutions for

cos (y) = \frac{6 - 3 2}{8}

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

y = arccos (\frac{6 - 3 2}{8}) + 36 0^{\circ} n, y = 36 0^{\circ} - arccos (\frac{6 - 3 2}{8}) + 36 0^{\circ} n

y = arccos (\frac{6 - 3 2}{8}) + 36 0^{\circ} n, y = 36 0^{\circ} - arccos (\frac{6 - 3 2}{8}) + 36 0^{\circ} n

Show solutions in decimal form

y = 1.34932 \dots + 36 0^{\circ} n, y = 36 0^{\circ} - 1.34932 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos(45)+4cos(y)=3 ?
The general solution for 3cos(45)+4cos(y)=3 is y=1.34932…+360n,y=360-1.34932…+360n