What is the general solution for 1/2 (6.50941)+cos(θ)(10.40478)=12 ?

The general solution for 1/2 (6.50941)+cos(θ)(10.40478)=12 is θ=0.57257…+2pin,θ=2pi-0.57257…+2pin

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Popular Trigonometry >

1/2 (6.50941)+cos(θ)(10.40478)=12

Solution

\frac{1}{2} (6.50941) + cos (θ) (10.40478) = 12

Solution

θ = 0.57257 \dots + 2 πn, θ = 2 π - 0.57257 \dots + 2 πn

Degrees

θ = 32.80625 \dots^{\circ} + 36 0^{\circ} n, θ = 327.19374 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{2} (6.50941) + cos (θ) (10.40478) = 12

\frac{1}{2} \cdot 6.50941 = \frac{6.50941}{2}

\frac{1}{2} \cdot 6.50941

Multiply fractions:

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

= \frac{1 \cdot 6.50941}{2}

Multiply the numbers:

1 \cdot 6.50941 = 6.50941

= \frac{6.50941}{2}

\frac{6.50941}{2} + 10.40478 cos (θ) = 12

Move

\frac{6.50941}{2}

to the right side

\frac{6.50941}{2} + 10.40478 cos (θ) = 12

Subtract

\frac{6.50941}{2}

from both sides

\frac{6.50941}{2} + 10.40478 cos (θ) - \frac{6.50941}{2} = 12 - \frac{6.50941}{2}

Simplify

\frac{6.50941}{2} + 10.40478 cos (θ) - \frac{6.50941}{2} = 12 - \frac{6.50941}{2}

Simplify

\frac{6.50941}{2} + 10.40478 cos (θ) - \frac{6.50941}{2} : 10.40478 cos (θ)

\frac{6.50941}{2} + 10.40478 cos (θ) - \frac{6.50941}{2}

Add similar elements:

\frac{6.50941}{2} - \frac{6.50941}{2} = 0

= 10.40478 cos (θ)

Simplify

12 - \frac{6.50941}{2} : \frac{17.49059}{2}

12 - \frac{6.50941}{2}

Convert element to fraction:

12 = \frac{12 \cdot 2}{2}

= \frac{12 \cdot 2}{2} - \frac{6.50941}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{12 \cdot 2 - 6.50941}{2}

12 \cdot 2 - 6.50941 = 17.49059

12 \cdot 2 - 6.50941

Multiply the numbers:

12 \cdot 2 = 24

= 24 - 6.50941

Subtract the numbers:

24 - 6.50941 = 17.49059

= 17.49059

= \frac{17.49059}{2}

10.40478 cos (θ) = \frac{17.49059}{2}

10.40478 cos (θ) = \frac{17.49059}{2}

10.40478 cos (θ) = \frac{17.49059}{2}

Divide both sides by

10.40478

10.40478 cos (θ) = \frac{17.49059}{2}

Divide both sides by

10.40478

\frac{10.40478 cos ( θ )}{10.40478} = \frac{\frac{17.49059}{2}}{10.40478}

Simplify

\frac{10.40478 cos ( θ )}{10.40478} = \frac{\frac{17.49059}{2}}{10.40478}

Simplify

\frac{10.40478 cos ( θ )}{10.40478} : cos (θ)

\frac{10.40478 cos ( θ )}{10.40478}

Cancel the common factor:

10.40478

= cos (θ)

Simplify

\frac{\frac{17.49059}{2}}{10.40478} : \frac{17.49059}{20.80956}

\frac{\frac{17.49059}{2}}{10.40478}

Apply the fraction rule:

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

= \frac{17.49059}{2 \cdot 10.40478}

Multiply the numbers:

2 \cdot 10.40478 = 20.80956

= \frac{17.49059}{20.80956}

cos (θ) = \frac{17.49059}{20.80956}

cos (θ) = \frac{17.49059}{20.80956}

cos (θ) = \frac{17.49059}{20.80956}

Apply trig inverse properties

cos (θ) = \frac{17.49059}{20.80956}

General solutions for

cos (θ) = \frac{17.49059}{20.80956}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

θ = arccos (\frac{17.49059}{20.80956}) + 2 πn, θ = 2 π - arccos (\frac{17.49059}{20.80956}) + 2 πn

θ = arccos (\frac{17.49059}{20.80956}) + 2 πn, θ = 2 π - arccos (\frac{17.49059}{20.80956}) + 2 πn

Show solutions in decimal form

θ = 0.57257 \dots + 2 πn, θ = 2 π - 0.57257 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 1/2 (6.50941)+cos(θ)(10.40478)=12 ?
The general solution for 1/2 (6.50941)+cos(θ)(10.40478)=12 is θ=0.57257…+2pin,θ=2pi-0.57257…+2pin