What is the general solution for sin(2v)=sin(16) ?

The general solution for sin(2v)=sin(16) is v=8+180n,v=90-8+180n

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sin(2v)=sin(16)

Solution

sin (2 v) = sin (1 6^{\circ})

Solution

v = 8^{\circ} + 18 0^{\circ} n, v = 9 0^{\circ} - 8^{\circ} + 18 0^{\circ} n

Radians

v = \frac{2 π}{45} + πn, v = \frac{π}{2} - \frac{2 π}{45} + πn

Solution steps

sin (2 v) = sin (1 6^{\circ})

Apply trig inverse properties

sin (2 v) = sin (1 6^{\circ})

General solutions for

sin (2 v) = sin (1 6^{\circ})

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

2 v = arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n, 2 v = 18 0^{\circ} - arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n

2 v = arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n, 2 v = 18 0^{\circ} - arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n

Solve

2 v = arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n : v = 8^{\circ} + 18 0^{\circ} n

2 v = arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n

Divide both sides by

2

2 v = arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n

Divide both sides by

2

\frac{2 v}{2} = \frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

Simplify

\frac{2 v}{2} = \frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

Simplify

\frac{2 v}{2} : v

\frac{2 v}{2}

Divide the numbers:

\frac{2}{2} = 1

= v

Simplify

\frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} : 8^{\circ} + 18 0^{\circ} n

\frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

\frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} = 8^{\circ}

\frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2}

arcsin (sin (1 6^{\circ})) = 1 6^{\circ}

arcsin (sin (1 6^{\circ}))

For

- 9 0^{\circ} \leq v \leq 9 0^{\circ}, a rcs in (s in (x)) = x

- 9 0^{\circ} \leq 1 6^{\circ} \leq 9 0^{\circ}

= 1 6^{\circ}

= \frac{1 6 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{72 0 ^{\circ}}{45 \cdot 2}

Multiply the numbers:

45 \cdot 2 = 90

= 8^{\circ}

Cancel the common factor:

2

= 8^{\circ}

= 8^{\circ} + \frac{36 0 ^{\circ} n}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

= 8^{\circ} + 18 0^{\circ} n

v = 8^{\circ} + 18 0^{\circ} n

v = 8^{\circ} + 18 0^{\circ} n

v = 8^{\circ} + 18 0^{\circ} n

Solve

2 v = 18 0^{\circ} - arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n : v = 9 0^{\circ} - 8^{\circ} + 18 0^{\circ} n

2 v = 18 0^{\circ} - arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n

Divide both sides by

2

2 v = 18 0^{\circ} - arcsin (sin (1 6^{\circ})) + 36 0^{\circ} n

Divide both sides by

2

\frac{2 v}{2} = 9 0^{\circ} - \frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

Simplify

\frac{2 v}{2} = 9 0^{\circ} - \frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

Simplify

\frac{2 v}{2} : v

\frac{2 v}{2}

Divide the numbers:

\frac{2}{2} = 1

= v

Simplify

9 0^{\circ} - \frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} : 9 0^{\circ} - 8^{\circ} + 18 0^{\circ} n

9 0^{\circ} - \frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

\frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2} = 8^{\circ}

\frac{arcsin ( sin ( 1 6 ^{\circ} ) )}{2}

arcsin (sin (1 6^{\circ})) = 1 6^{\circ}

arcsin (sin (1 6^{\circ}))

For

- 9 0^{\circ} \leq v \leq 9 0^{\circ}, a rcs in (s in (x)) = x

- 9 0^{\circ} \leq 1 6^{\circ} \leq 9 0^{\circ}

= 1 6^{\circ}

= \frac{1 6 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{72 0 ^{\circ}}{45 \cdot 2}

Multiply the numbers:

45 \cdot 2 = 90

= 8^{\circ}

Cancel the common factor:

2

= 8^{\circ}

= 9 0^{\circ} - 8^{\circ} + \frac{36 0 ^{\circ} n}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

= 9 0^{\circ} - 8^{\circ} + 18 0^{\circ} n

v = 9 0^{\circ} - 8^{\circ} + 18 0^{\circ} n

v = 9 0^{\circ} - 8^{\circ} + 18 0^{\circ} n

v = 9 0^{\circ} - 8^{\circ} + 18 0^{\circ} n

v = 8^{\circ} + 18 0^{\circ} n, v = 9 0^{\circ} - 8^{\circ} + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2v)=sin(16) ?
The general solution for sin(2v)=sin(16) is v=8+180n,v=90-8+180n