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Popular >

sin(3x-50)cos(x)-cos(3x-50)sin(x)=0,0<= x<= 180

Solution

sin (3 x - 50) cos (x) - cos (3 x - 50) sin (x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

Solution

No Solution for x \in R

Solution steps

sin (3 x - 50) cos (x) - cos (3 x - 50) sin (x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

Rewrite using trig identities

sin (3 x - 50) cos (x) - cos (3 x - 50) sin (x)

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (3 x - 50 - x)

sin (3 x - 50 - x) = 0

General solutions for

sin (3 x - 50 - x) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

3 x - 50 - x = 0 + 36 0^{\circ} n, 3 x - 50 - x = 18 0^{\circ} + 36 0^{\circ} n

3 x - 50 - x = 0 + 36 0^{\circ} n, 3 x - 50 - x = 18 0^{\circ} + 36 0^{\circ} n

Solve

3 x - 50 - x = 0 + 36 0^{\circ} n : x = 18 0^{\circ} n + 25

3 x - 50 - x = 0 + 36 0^{\circ} n

Group like terms

3 x - x - 50 = 0 + 36 0^{\circ} n

Add similar elements:

3 x - x = 2 x

2 x - 50 = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

2 x - 50 = 36 0^{\circ} n

Move

50

to the right side

2 x - 50 = 36 0^{\circ} n

Add

50

to both sides

2 x - 50 + 50 = 36 0^{\circ} n + 50

Simplify

2 x = 36 0^{\circ} n + 50

2 x = 36 0^{\circ} n + 50

Divide both sides by

2

2 x = 36 0^{\circ} n + 50

Divide both sides by

2

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{50}{2}

Simplify

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{50}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{2} + \frac{50}{2} : 18 0^{\circ} n + 25

\frac{36 0 ^{\circ} n}{2} + \frac{50}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n + \frac{50}{2}

Divide the numbers:

\frac{50}{2} = 25

= 18 0^{\circ} n + 25

x = 18 0^{\circ} n + 25

x = 18 0^{\circ} n + 25

x = 18 0^{\circ} n + 25

Solve

3 x - 50 - x = 18 0^{\circ} + 36 0^{\circ} n : x = 9 0^{\circ} + 25 + 18 0^{\circ} n

3 x - 50 - x = 18 0^{\circ} + 36 0^{\circ} n

Group like terms

3 x - x - 50 = 18 0^{\circ} + 36 0^{\circ} n

Add similar elements:

3 x - x = 2 x

2 x - 50 = 18 0^{\circ} + 36 0^{\circ} n

Move

50

to the right side

2 x - 50 = 18 0^{\circ} + 36 0^{\circ} n

Add

50

to both sides

2 x - 50 + 50 = 18 0^{\circ} + 36 0^{\circ} n + 50

Simplify

2 x = 18 0^{\circ} + 36 0^{\circ} n + 50

2 x = 18 0^{\circ} + 36 0^{\circ} n + 50

Divide both sides by

2

2 x = 18 0^{\circ} + 36 0^{\circ} n + 50

Divide both sides by

2

\frac{2 x}{2} = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} + \frac{50}{2}

Simplify

\frac{2 x}{2} = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} + \frac{50}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} + \frac{50}{2} : 9 0^{\circ} + 25 + 18 0^{\circ} n

9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} + \frac{50}{2}

Group like terms

= 9 0^{\circ} + \frac{50}{2} + \frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{50}{2} = 25

= 9 0^{\circ} + 25 + \frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 9 0^{\circ} + 25 + 18 0^{\circ} n

x = 9 0^{\circ} + 25 + 18 0^{\circ} n

x = 9 0^{\circ} + 25 + 18 0^{\circ} n

x = 9 0^{\circ} + 25 + 18 0^{\circ} n

x = 18 0^{\circ} n + 25, x = 9 0^{\circ} + 25 + 18 0^{\circ} n

Solutions for the range

0 \leq x \leq 18 0^{\circ}

No Solution for x \in R

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