What is the general solution for 300= 1/32 100^2sin(2x) ?

The general solution for 300= 1/32 100^2sin(2x) is x=(1.28700…)/2+pin,x= pi/2-(1.28700…)/2+pin

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300= 1/32 100^2sin(2x)

Solution

300 = \frac{1}{32} 10 0^{2} sin (2 x)

Solution

x = \frac{1.28700 \dots}{2} + πn, x = \frac{π}{2} - \frac{1.28700 \dots}{2} + πn

Degrees

x = 36.86989 \dots^{\circ} + 18 0^{\circ} n, x = 53.13010 \dots^{\circ} + 18 0^{\circ} n

Solution steps

300 = \frac{1}{32} \cdot 10 0^{2} sin (2 x)

Switch sides

\frac{1}{32} \cdot 10 0^{2} sin (2 x) = 300

Simplify

\frac{1}{32} \cdot 10 0^{2} : \frac{625}{2}

\frac{1}{32} \cdot 10 0^{2}

Multiply fractions:

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

= \frac{1 \cdot 10 0 ^{2}}{32}

Multiply:

1 \cdot 10 0^{2} = 10 0^{2}

= \frac{10 0 ^{2}}{32}

Factor

10 0^{2} : 2^{4} \cdot 5^{4}

Factor

100 = 2^{2} \cdot 5^{2}

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

Apply exponent rule:

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

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

Simplify

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

(2^{2})^{2}

Apply exponent rule:

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

= 2^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= 2^{4}

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^{4} \cdot 5^{4}

Factor

32 : 2^{5}

Factor

32 = 2^{5}

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

Cancel

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{4}}{2 ^{5}} = \frac{1}{2 ^{5 - 4}}

= \frac{5 ^{4}}{2 ^{5 - 4}}

Subtract the numbers:

5 - 4 = 1

= \frac{5 ^{4}}{2}

= \frac{5 ^{4}}{2}

5^{4} = 625

= \frac{625}{2}

\frac{625}{2} sin (2 x) = 300

Multiply both sides by

2

\frac{625}{2} sin (2 x) = 300

Multiply both sides by

2

2 \cdot \frac{625}{2} sin (2 x) = 300 \cdot 2

Simplify

625 sin (2 x) = 600

625 sin (2 x) = 600

Divide both sides by

625

625 sin (2 x) = 600

Divide both sides by

625

\frac{625 sin ( 2 x )}{625} = \frac{600}{625}

Simplify

sin (2 x) = \frac{24}{25}

sin (2 x) = \frac{24}{25}

Apply trig inverse properties

sin (2 x) = \frac{24}{25}

General solutions for

sin (2 x) = \frac{24}{25}

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

2 x = arcsin (\frac{24}{25}) + 2 πn, 2 x = π - arcsin (\frac{24}{25}) + 2 πn

2 x = arcsin (\frac{24}{25}) + 2 πn, 2 x = π - arcsin (\frac{24}{25}) + 2 πn

Solve

2 x = arcsin (\frac{24}{25}) + 2 πn : x = \frac{arcsin ( \frac{24}{25} )}{2} + πn

2 x = arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

2 x = arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsin ( \frac{24}{25} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arcsin ( \frac{24}{25} )}{2} + πn

x = \frac{arcsin ( \frac{24}{25} )}{2} + πn

Solve

2 x = π - arcsin (\frac{24}{25}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{24}{25} )}{2} + πn

2 x = π - arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

2 x = π - arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{24}{25} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} - \frac{arcsin ( \frac{24}{25} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{24}{25} )}{2} + πn

x = \frac{arcsin ( \frac{24}{25} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{24}{25} )}{2} + πn

Show solutions in decimal form

x = \frac{1.28700 \dots}{2} + πn, x = \frac{π}{2} - \frac{1.28700 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 300= 1/32 100^2sin(2x) ?
The general solution for 300= 1/32 100^2sin(2x) is x=(1.28700…)/2+pin,x= pi/2-(1.28700…)/2+pin