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1000cos(45)=520cos(35)+1043.937687cos(θ)

Solution

1000 cos (4 5^{\circ}) = 520 cos (3 5^{\circ}) + 1043.937687 cos (θ)

θ = 1.29811 \dots + 36 0^{\circ} n, θ = 36 0^{\circ} - 1.29811 \dots + 36 0^{\circ} n

Radians

θ = 1.29811 \dots + 2 πn, θ = 2 π - 1.29811 \dots + 2 πn

Solution steps

1000 cos (4 5^{\circ}) = 520 cos (3 5^{\circ}) + 1043.937687 cos (θ)

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}

1000 \cdot \frac{2}{2} = 520 cos (3 5^{\circ}) + 1043.937687 cos (θ)

Switch sides

520 cos (3 5^{\circ}) + 1043.937687 cos (θ) = 1000 \cdot \frac{2}{2}

Multiply fractions:

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

520 cos (3 5^{\circ}) + 1043.937687 cos (θ) = \frac{2 \cdot 1000}{2}

Divide the numbers:

\frac{1000}{2} = 500

520 cos (3 5^{\circ}) + 1043.937687 cos (θ) = 5002

Move

520 cos (3 5^{\circ})

to the right side

520 cos (3 5^{\circ}) + 1043.937687 cos (θ) = 5002

Subtract

520 cos (3 5^{\circ})

from both sides

520 cos (3 5^{\circ}) + 1043.937687 cos (θ) - 520 cos (3 5^{\circ}) = 5002 - 520 cos (3 5^{\circ})

Simplify

1043.937687 cos (θ) = 5002 - 520 cos (3 5^{\circ})

1043.937687 cos (θ) = 5002 - 520 cos (3 5^{\circ})

Divide both sides by

1043.937687

1043.937687 cos (θ) = 5002 - 520 cos (3 5^{\circ})

Divide both sides by

1043.937687

\frac{1043.937687 cos ( θ )}{1043.937687} = \frac{500 2}{1043.937687} - \frac{520 cos ( 3 5 ^{\circ} )}{1043.937687}

Simplify

\frac{1043.937687 cos ( θ )}{1043.937687} = \frac{500 2}{1043.937687} - \frac{520 cos ( 3 5 ^{\circ} )}{1043.937687}

Simplify

\frac{1043.937687 cos ( θ )}{1043.937687} : cos (θ)

\frac{1043.937687 cos ( θ )}{1043.937687}

Cancel the common factor:

1043.937687

= cos (θ)

Simplify

\frac{500 2}{1043.937687} - \frac{520 cos ( 3 5 ^{\circ} )}{1043.937687} : \frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}

\frac{500 2}{1043.937687} - \frac{520 cos ( 3 5 ^{\circ} )}{1043.937687}

Apply rule

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

= \frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}

cos (θ) = \frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}

cos (θ) = \frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}

cos (θ) = \frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}

Apply trig inverse properties

cos (θ) = \frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}

General solutions for

cos (θ) = \frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}

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

θ = arccos (\frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}) + 36 0^{\circ} n, θ = 36 0^{\circ} - arccos (\frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}) + 36 0^{\circ} n

θ = arccos (\frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}) + 36 0^{\circ} n, θ = 36 0^{\circ} - arccos (\frac{500 2 - 520 cos ( 3 5 ^{\circ} )}{1043.937687}) + 36 0^{\circ} n

Show solutions in decimal form

θ = 1.29811 \dots + 36 0^{\circ} n, θ = 36 0^{\circ} - 1.29811 \dots + 36 0^{\circ} n

4 sin (\frac{x}{2}) = 4 cos (\frac{x}{2})

tan^{2} (θ) + 8 tan (θ) + 9 = 0

2 tan^{2} (x) - 1 = 3 tan (x)

8 csc^{2} (x) - 2 cot (x) = 9

cos (d) = \frac{9}{15}