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Popular Trigonometry >

solvefor n,sin(x)+sin(13 n/2-x)=1

Solution

solvefor

Solution

n = \frac{2 arcsin ( 1 - sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}, n = \frac{2 π}{13} + \frac{2 arcsin ( - 1 + sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

Solution steps

sin (x) + sin (13 \cdot \frac{n}{2} - x) = 1

Move

sin (x)

to the right side

sin (x) + sin (13 \frac{n}{2} - x) = 1

Subtract

sin (x)

from both sides

sin (x) + sin (13 \frac{n}{2} - x) - sin (x) = 1 - sin (x)

Simplify

sin (13 \frac{n}{2} - x) = 1 - sin (x)

sin (13 \frac{n}{2} - x) = 1 - sin (x)

Apply trig inverse properties

sin (13 \cdot \frac{n}{2} - x) = 1 - sin (x)

General solutions for

sin (13 \frac{n}{2} - x) = 1 - sin (x)

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

13 \cdot \frac{n}{2} - x = arcsin (1 - sin (x)) + 2 πk, 13 \cdot \frac{n}{2} - x = π + arcsin (- 1 + sin (x)) + 2 πk

13 \cdot \frac{n}{2} - x = arcsin (1 - sin (x)) + 2 πk, 13 \cdot \frac{n}{2} - x = π + arcsin (- 1 + sin (x)) + 2 πk

Solve

13 \cdot \frac{n}{2} - x = arcsin (1 - sin (x)) + 2 πk : n = \frac{2 arcsin ( 1 - sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

13 \cdot \frac{n}{2} - x = arcsin (1 - sin (x)) + 2 πk

Move

x

to the right side

13 \cdot \frac{n}{2} - x = arcsin (1 - sin (x)) + 2 πk

Add

x

to both sides

13 \cdot \frac{n}{2} - x + x = arcsin (1 - sin (x)) + 2 πk + x

Simplify

13 \cdot \frac{n}{2} = arcsin (1 - sin (x)) + 2 πk + x

13 \cdot \frac{n}{2} = arcsin (1 - sin (x)) + 2 πk + x

Refine

13 \cdot \frac{n}{2} : \frac{13 n}{2}

13 \cdot \frac{n}{2}

Multiply fractions:

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

= \frac{n \cdot 13}{2}

\frac{13 n}{2} = arcsin (1 - sin (x)) + 2 πk + x

Multiply both sides by

2

\frac{13 n}{2} = arcsin (1 - sin (x)) + 2 πk + x

Multiply both sides by

2

\frac{13 n}{2} \cdot 2 = arcsin (1 - sin (x)) \cdot 2 + 2 πk \cdot 2 + x \cdot 2

Simplify

\frac{13 n}{2} \cdot 2 = arcsin (1 - sin (x)) \cdot 2 + 2 πk \cdot 2 + x \cdot 2

Simplify

\frac{13 n}{2} \cdot 2 : 13 n

\frac{13 n}{2} \cdot 2

Multiply fractions:

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

= \frac{13 n \cdot 2}{2}

Cancel the common factor:

2

= 13 n

Simplify

arcsin (1 - sin (x)) \cdot 2 : 2 arcsin (1 - sin (x))

arcsin (1 - sin (x)) \cdot 2

Apply the commutative law:

arcsin (1 - sin (x)) \cdot 2 = 2 arcsin (1 - sin (x))

2 arcsin (1 - sin (x))

Simplify

2 πk \cdot 2 : 4 πk

2 πk \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πk

Simplify

x \cdot 2 : 2 x

x \cdot 2

Apply the commutative law:

x \cdot 2 = 2 x

2 x

13 n = 2 arcsin (1 - sin (x)) + 4 πk + 2 x

13 n = 2 arcsin (1 - sin (x)) + 4 πk + 2 x

13 n = 2 arcsin (1 - sin (x)) + 4 πk + 2 x

Divide both sides by

13

13 n = 2 arcsin (1 - sin (x)) + 4 πk + 2 x

Divide both sides by

13

\frac{13 n}{13} = \frac{2 arcsin ( 1 - sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

Simplify

n = \frac{2 arcsin ( 1 - sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

n = \frac{2 arcsin ( 1 - sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

Solve

13 \cdot \frac{n}{2} - x = π + arcsin (- 1 + sin (x)) + 2 πk : n = \frac{2 π}{13} + \frac{2 arcsin ( - 1 + sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

13 \cdot \frac{n}{2} - x = π + arcsin (- 1 + sin (x)) + 2 πk

Move

x

to the right side

13 \cdot \frac{n}{2} - x = π + arcsin (- 1 + sin (x)) + 2 πk

Add

x

to both sides

13 \cdot \frac{n}{2} - x + x = π + arcsin (- 1 + sin (x)) + 2 πk + x

Simplify

13 \cdot \frac{n}{2} = π + arcsin (- 1 + sin (x)) + 2 πk + x

13 \cdot \frac{n}{2} = π + arcsin (- 1 + sin (x)) + 2 πk + x

Refine

13 \cdot \frac{n}{2} : \frac{13 n}{2}

13 \cdot \frac{n}{2}

Multiply fractions:

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

= \frac{n \cdot 13}{2}

\frac{13 n}{2} = π + arcsin (- 1 + sin (x)) + 2 πk + x

Multiply both sides by

2

\frac{13 n}{2} = π + arcsin (- 1 + sin (x)) + 2 πk + x

Multiply both sides by

2

\frac{13 n}{2} \cdot 2 = π 2 + arcsin (- 1 + sin (x)) \cdot 2 + 2 πk \cdot 2 + x \cdot 2

Simplify

\frac{13 n}{2} \cdot 2 = π 2 + arcsin (- 1 + sin (x)) \cdot 2 + 2 πk \cdot 2 + x \cdot 2

Simplify

\frac{13 n}{2} \cdot 2 : 13 n

\frac{13 n}{2} \cdot 2

Multiply fractions:

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

= \frac{13 n \cdot 2}{2}

Cancel the common factor:

2

= 13 n

Simplify

π 2 : 2 π

π 2

Apply the commutative law:

π 2 = 2 π

2 π

Simplify

arcsin (- 1 + sin (x)) \cdot 2 : 2 arcsin (- 1 + sin (x))

arcsin (- 1 + sin (x)) \cdot 2

Apply the commutative law:

arcsin (- 1 + sin (x)) \cdot 2 = 2 arcsin (- 1 + sin (x))

2 arcsin (- 1 + sin (x))

Simplify

2 πk \cdot 2 : 4 πk

2 πk \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πk

Simplify

x \cdot 2 : 2 x

x \cdot 2

Apply the commutative law:

x \cdot 2 = 2 x

2 x

13 n = 2 π + 2 arcsin (- 1 + sin (x)) + 4 πk + 2 x

13 n = 2 π + 2 arcsin (- 1 + sin (x)) + 4 πk + 2 x

13 n = 2 π + 2 arcsin (- 1 + sin (x)) + 4 πk + 2 x

Divide both sides by

13

13 n = 2 π + 2 arcsin (- 1 + sin (x)) + 4 πk + 2 x

Divide both sides by

13

\frac{13 n}{13} = \frac{2 π}{13} + \frac{2 arcsin ( - 1 + sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

Simplify

n = \frac{2 π}{13} + \frac{2 arcsin ( - 1 + sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

n = \frac{2 π}{13} + \frac{2 arcsin ( - 1 + sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

n = \frac{2 arcsin ( 1 - sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}, n = \frac{2 π}{13} + \frac{2 arcsin ( - 1 + sin ( x ) )}{13} + \frac{4 πk}{13} + \frac{2 x}{13}

Graph

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