What is the general solution for 1/32*1200^2*sin(2θ)=3000 ?

The general solution for 1/32*1200^2*sin(2θ)=3000 is θ=(0.06671…)/2+pin,θ= pi/2-(0.06671…)/2+pin

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1/321200^2sin(2θ)=3000

Solution

\frac{1}{32} \cdot 120 0^{2} \cdot sin (2 θ) = 3000

Solution

θ = \frac{0.06671 \dots}{2} + πn, θ = \frac{π}{2} - \frac{0.06671 \dots}{2} + πn

Radians

θ = 0.03335 \dots + πn, θ = \frac{π}{2} - \frac{0.06671 \dots}{2} + πn

Solution steps

\frac{1}{32} \cdot 120 0^{2} sin (2 θ) = 3000

Simplify

\frac{1}{32} \cdot 120 0^{2} : 45000

\frac{1}{32} \cdot 120 0^{2}

Multiply fractions:

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

= \frac{1 \cdot 120 0 ^{2}}{32}

Multiply:

1 \cdot 120 0^{2} = 120 0^{2}

= \frac{120 0 ^{2}}{32}

Factor

120 0^{2} : 2^{8} \cdot 5^{4} \cdot 3^{2}

Factor

1200 = 2^{4} \cdot 5^{2} \cdot 3

= (2^{4} \cdot 5^{2} \cdot 3)^{2}

Apply exponent rule:

(ab)^{c} = a^{c} b^{c}

= 3^{2} (2^{4})^{2} (5^{2})^{2}

Simplify

(2^{4})^{2} : 2^{8}

(2^{4})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= 2^{8}

Simplify

(5^{2})^{2} : 5^{4}

(5^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 5^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= 5^{4}

= 2^{8} \cdot 5^{4} \cdot 3^{2}

Factor

32 : 2^{5}

Factor

32 = 2^{5}

= \frac{2 ^{8} \cdot 5 ^{4} \cdot 3 ^{2}}{2 ^{5}}

Cancel

\frac{2 ^{8} \cdot 3 ^{2} \cdot 5 ^{4}}{2 ^{5}} : 5^{4} \cdot 2^{3} \cdot 3^{2}

\frac{2 ^{8} \cdot 3 ^{2} \cdot 5 ^{4}}{2 ^{5}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{8}}{2 ^{5}} = 2^{8 - 5}

= 5^{4} \cdot 3^{2} \cdot 2^{8 - 5}

Subtract the numbers:

8 - 5 = 3

= 5^{4} \cdot 2^{3} \cdot 3^{2}

= 5^{4} \cdot 2^{3} \cdot 3^{2}

5^{4} = 625

= 2^{3} \cdot 3^{2} \cdot 625

2^{3} = 8

= 3^{2} \cdot 625 \cdot 8

3^{2} = 9

= 625 \cdot 8 \cdot 9

Multiply the numbers:

625 \cdot 8 \cdot 9 = 45000

= 45000

45000 sin (2 θ) = 3000

Divide both sides by

45000

45000 sin (2 θ) = 3000

Divide both sides by

45000

\frac{45000 sin ( 2 θ )}{45000} = \frac{3000}{45000}

Simplify

sin (2 θ) = \frac{1}{15}

sin (2 θ) = \frac{1}{15}

Apply trig inverse properties

sin (2 θ) = \frac{1}{15}

General solutions for

sin (2 θ) = \frac{1}{15}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 θ = arcsin (\frac{1}{15}) + 2 πn, 2 θ = π - arcsin (\frac{1}{15}) + 2 πn

2 θ = arcsin (\frac{1}{15}) + 2 πn, 2 θ = π - arcsin (\frac{1}{15}) + 2 πn

Solve

2 θ = arcsin (\frac{1}{15}) + 2 πn : θ = \frac{arcsin ( \frac{1}{15} )}{2} + πn

2 θ = arcsin (\frac{1}{15}) + 2 πn

Divide both sides by

2

2 θ = arcsin (\frac{1}{15}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{arcsin ( \frac{1}{15} )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{arcsin ( \frac{1}{15} )}{2} + πn

θ = \frac{arcsin ( \frac{1}{15} )}{2} + πn

Solve

2 θ = π - arcsin (\frac{1}{15}) + 2 πn : θ = \frac{π}{2} - \frac{arcsin ( \frac{1}{15} )}{2} + πn

2 θ = π - arcsin (\frac{1}{15}) + 2 πn

Divide both sides by

2

2 θ = π - arcsin (\frac{1}{15}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{15} )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} - \frac{arcsin ( \frac{1}{15} )}{2} + πn

θ = \frac{π}{2} - \frac{arcsin ( \frac{1}{15} )}{2} + πn

θ = \frac{arcsin ( \frac{1}{15} )}{2} + πn, θ = \frac{π}{2} - \frac{arcsin ( \frac{1}{15} )}{2} + πn

Show solutions in decimal form

θ = \frac{0.06671 \dots}{2} + πn, θ = \frac{π}{2} - \frac{0.06671 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 1/32*1200^2*sin(2θ)=3000 ?
The general solution for 1/32*1200^2*sin(2θ)=3000 is θ=(0.06671…)/2+pin,θ= pi/2-(0.06671…)/2+pin