What is the general solution for 100=50sqrt(40)cos(θ) ?

The general solution for 100=50sqrt(40)cos(θ) is θ=1.24904…+2pin,θ=2pi-1.24904…+2pin

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Popular Trigonometry >

100=50sqrt(40)cos(θ)

Solution

100 = 5040 cos (θ)

Solution

θ = 1.24904 \dots + 2 πn, θ = 2 π - 1.24904 \dots + 2 πn

Degrees

θ = 71.56505 \dots^{\circ} + 36 0^{\circ} n, θ = 288.43494 \dots^{\circ} + 36 0^{\circ} n

Solution steps

100 = 5040 cos (θ)

Switch sides

5040 cos (θ) = 100

Simplify

5040 : 10010

5040

40 = 210

40

Prime factorization of

40 : 2^{3} \cdot 5

40

40

divides by

240 = 20 \cdot 2

= 2 \cdot 20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 5

= 2^{3} \cdot 5

= 2^{3} \cdot 5

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2 \cdot 5

Apply radical rule:

= 2^{2} 2 \cdot 5

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 5

Refine

= 210

= 50 \cdot 210

Multiply the numbers:

50 \cdot 2 = 100

= 10010

10010 cos (θ) = 100

Divide both sides by

10010

10010 cos (θ) = 100

Divide both sides by

10010

\frac{100 10 cos ( θ )}{100 10} = \frac{100}{100 10}

Simplify

\frac{100 10 cos ( θ )}{100 10} = \frac{100}{100 10}

Simplify

\frac{100 10 cos ( θ )}{100 10} : cos (θ)

\frac{100 10 cos ( θ )}{100 10}

Divide the numbers:

\frac{100}{100} = 1

= \frac{10 cos ( θ )}{10}

Cancel the common factor:

10

= cos (θ)

Simplify

\frac{100}{100 10} : \frac{10}{10}

\frac{100}{100 10}

Divide the numbers:

\frac{100}{100} = 1

= \frac{1}{10}

Rationalize

\frac{1}{10} : \frac{10}{10}

\frac{1}{10}

Multiply by the conjugate

\frac{10}{10}

= \frac{1 \cdot 10}{10 10}

1 \cdot 10 = 10

1010 = 10

1010

Apply radical rule:

a a = a

1010 = 10

= 10

= \frac{10}{10}

= \frac{10}{10}

cos (θ) = \frac{10}{10}

cos (θ) = \frac{10}{10}

cos (θ) = \frac{10}{10}

Apply trig inverse properties

cos (θ) = \frac{10}{10}

General solutions for

cos (θ) = \frac{10}{10}

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

θ = arccos (\frac{10}{10}) + 2 πn, θ = 2 π - arccos (\frac{10}{10}) + 2 πn

θ = arccos (\frac{10}{10}) + 2 πn, θ = 2 π - arccos (\frac{10}{10}) + 2 πn

Show solutions in decimal form

θ = 1.24904 \dots + 2 πn, θ = 2 π - 1.24904 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 100=50sqrt(40)cos(θ) ?
The general solution for 100=50sqrt(40)cos(θ) is θ=1.24904…+2pin,θ=2pi-1.24904…+2pin