What is the general solution for 885cos(θ)-70=cos(250)+250 ?

The general solution for 885cos(θ)-70=cos(250)+250 is θ=1.48705…+360n,θ=360-1.48705…+360n

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Popular Trigonometry >

885cos(θ)-70=cos(250)+250

Solution

885 cos (θ) - 70 = cos (25 0^{\circ}) + 25 0^{\circ}

Solution

θ = 1.48705 \dots + 36 0^{\circ} n, θ = 36 0^{\circ} - 1.48705 \dots + 36 0^{\circ} n

Radians

θ = 1.48705 \dots + 2 πn, θ = 2 π - 1.48705 \dots + 2 πn

Solution steps

885 cos (θ) - 70 = cos (25 0^{\circ}) + 25 0^{\circ}

Move

70

to the right side

885 cos (θ) - 70 = cos (25 0^{\circ}) + 25 0^{\circ}

Add

70

to both sides

885 cos (θ) - 70 + 70 = cos (25 0^{\circ}) + 25 0^{\circ} + 70

Simplify

885 cos (θ) = cos (25 0^{\circ}) + 25 0^{\circ} + 70

885 cos (θ) = cos (25 0^{\circ}) + 25 0^{\circ} + 70

Divide both sides by

885

885 cos (θ) = cos (25 0^{\circ}) + 25 0^{\circ} + 70

Divide both sides by

885

\frac{885 cos ( θ )}{885} = \frac{cos ( 25 0 ^{\circ} )}{885} + \frac{25 0 ^{\circ}}{885} + \frac{70}{885}

Simplify

\frac{885 cos ( θ )}{885} = \frac{cos ( 25 0 ^{\circ} )}{885} + \frac{25 0 ^{\circ}}{885} + \frac{70}{885}

Simplify

\frac{885 cos ( θ )}{885} : cos (θ)

\frac{885 cos ( θ )}{885}

Divide the numbers:

\frac{885}{885} = 1

= cos (θ)

Simplify

\frac{cos ( 25 0 ^{\circ} )}{885} + \frac{25 0 ^{\circ}}{885} + \frac{70}{885} : \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}

\frac{cos ( 25 0 ^{\circ} )}{885} + \frac{25 0 ^{\circ}}{885} + \frac{70}{885}

Apply rule

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

= \frac{cos ( 25 0 ^{\circ} ) + 25 0 ^{\circ} + 70}{885}

Join

cos (25 0^{\circ}) + 25 0^{\circ} + 70 : \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{18}

cos (25 0^{\circ}) + 25 0^{\circ} + 70

Convert element to fraction:

cos (25 0^{\circ}) = \frac{cos ( 25 0 ^{\circ} ) 18}{18}, 70 = \frac{70 \cdot 18}{18}

= \frac{cos ( 25 0 ^{\circ} ) \cdot 18}{18} + 25 0^{\circ} + \frac{70 \cdot 18}{18}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( 25 0 ^{\circ} ) \cdot 18 + 450 0 ^{\circ} + 70 \cdot 18}{18}

Multiply the numbers:

70 \cdot 18 = 1260

= \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{18}

= \frac{\frac{18 c o s ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{18}}{885}

Apply the fraction rule:

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

= \frac{cos ( 25 0 ^{\circ} ) \cdot 18 + 450 0 ^{\circ} + 1260}{18 \cdot 885}

Multiply the numbers:

18 \cdot 885 = 15930

= \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}

cos (θ) = \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}

cos (θ) = \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}

cos (θ) = \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}

Apply trig inverse properties

cos (θ) = \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}

General solutions for

cos (θ) = \frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}

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

θ = arccos (\frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}) + 36 0^{\circ} n, θ = 36 0^{\circ} - arccos (\frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}) + 36 0^{\circ} n

θ = arccos (\frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}) + 36 0^{\circ} n, θ = 36 0^{\circ} - arccos (\frac{18 cos ( 25 0 ^{\circ} ) + 450 0 ^{\circ} + 1260}{15930}) + 36 0^{\circ} n

Show solutions in decimal form

θ = 1.48705 \dots + 36 0^{\circ} n, θ = 36 0^{\circ} - 1.48705 \dots + 36 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 885cos(θ)-70=cos(250)+250 ?
The general solution for 885cos(θ)-70=cos(250)+250 is θ=1.48705…+360n,θ=360-1.48705…+360n