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Popular Trigonometry >

sin(4y)+sin(6y)+cos(y)=0,0<= y<= pi/2

Solution

sin (4 y) + sin (6 y) + cos (y) = 0, 0 \leq y \leq \frac{π}{2}

Solution

y = \frac{π}{2}, y = \frac{7 π}{30}, y = \frac{11 π}{30}

Degrees

y = 9 0^{\circ}, y = 4 2^{\circ}, y = 6 6^{\circ}

Solution steps

sin (4 y) + sin (6 y) + cos (y) = 0, 0 \leq y \leq \frac{π}{2}

Rewrite using trig identities

cos (y) + sin (4 y) + sin (6 y)

Use the Sum to Product identity:

sin (s) + sin (t) = 2 sin (\frac{s + t}{2}) cos (\frac{s - t}{2})

= cos (y) + 2 sin (\frac{4 y + 6 y}{2}) cos (\frac{4 y - 6 y}{2})

2 sin (\frac{4 y + 6 y}{2}) cos (\frac{4 y - 6 y}{2}) = 2 cos (y) sin (5 y)

2 sin (\frac{4 y + 6 y}{2}) cos (\frac{4 y - 6 y}{2})

\frac{4 y + 6 y}{2} = 5 y

\frac{4 y + 6 y}{2}

Add similar elements:

4 y + 6 y = 10 y

= \frac{10 y}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 y

= 2 sin (5 y) cos (\frac{4 y - 6 y}{2})

\frac{4 y - 6 y}{2} = - y

\frac{4 y - 6 y}{2}

Add similar elements:

4 y - 6 y = - 2 y

= \frac{- 2 y}{2}

Apply the fraction rule:

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

= - \frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= - y

= 2 sin (5 y) cos (- y)

Use the negative angle identity:

cos (- x) = cos (x)

= 2 cos (y) sin (5 y)

= cos (y) + 2 cos (y) sin (5 y)

cos (y) + 2 cos (y) sin (5 y) = 0

Factor

cos (y) + 2 cos (y) sin (5 y) : cos (y) (2 sin (5 y) + 1)

cos (y) + 2 cos (y) sin (5 y)

Factor out common term

cos (y)

= cos (y) (1 + 2 sin (5 y))

cos (y) (2 sin (5 y) + 1) = 0

Solving each part separately

cos (y) = 0 or 2 sin (5 y) + 1 = 0

cos (y) = 0, 0 \leq y \leq \frac{π}{2} : y = \frac{π}{2}

cos (y) = 0, 0 \leq y \leq \frac{π}{2}

General solutions for

cos (y) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

y = \frac{π}{2} + 2 πn, y = \frac{3 π}{2} + 2 πn

y = \frac{π}{2} + 2 πn, y = \frac{3 π}{2} + 2 πn

Solutions for the range

0 \leq y \leq \frac{π}{2}

y = \frac{π}{2}

2 sin (5 y) + 1 = 0, 0 \leq y \leq \frac{π}{2} : y = \frac{7 π}{30}, y = \frac{11 π}{30}

2 sin (5 y) + 1 = 0, 0 \leq y \leq \frac{π}{2}

Move

1

to the right side

2 sin (5 y) + 1 = 0

Subtract

1

from both sides

2 sin (5 y) + 1 - 1 = 0 - 1

Simplify

2 sin (5 y) = - 1

2 sin (5 y) = - 1

Divide both sides by

2

2 sin (5 y) = - 1

Divide both sides by

2

\frac{2 sin ( 5 y )}{2} = \frac{- 1}{2}

Simplify

sin (5 y) = - \frac{1}{2}

sin (5 y) = - \frac{1}{2}

General solutions for

sin (5 y) = - \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

5 y = \frac{7 π}{6} + 2 πn, 5 y = \frac{11 π}{6} + 2 πn

5 y = \frac{7 π}{6} + 2 πn, 5 y = \frac{11 π}{6} + 2 πn

Solve

5 y = \frac{7 π}{6} + 2 πn : y = \frac{7 π}{30} + \frac{2 πn}{5}

5 y = \frac{7 π}{6} + 2 πn

Divide both sides by

5

5 y = \frac{7 π}{6} + 2 πn

Divide both sides by

5

\frac{5 y}{5} = \frac{\frac{7 π}{6}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 y}{5} = \frac{\frac{7 π}{6}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 y}{5} : y

\frac{5 y}{5}

Divide the numbers:

\frac{5}{5} = 1

= y

Simplify

\frac{\frac{7 π}{6}}{5} + \frac{2 πn}{5} : \frac{7 π}{30} + \frac{2 πn}{5}

\frac{\frac{7 π}{6}}{5} + \frac{2 πn}{5}

\frac{\frac{7 π}{6}}{5} = \frac{7 π}{30}

\frac{\frac{7 π}{6}}{5}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 5}

Multiply the numbers:

6 \cdot 5 = 30

= \frac{7 π}{30}

= \frac{7 π}{30} + \frac{2 πn}{5}

y = \frac{7 π}{30} + \frac{2 πn}{5}

y = \frac{7 π}{30} + \frac{2 πn}{5}

y = \frac{7 π}{30} + \frac{2 πn}{5}

Solve

5 y = \frac{11 π}{6} + 2 πn : y = \frac{11 π}{30} + \frac{2 πn}{5}

5 y = \frac{11 π}{6} + 2 πn

Divide both sides by

5

5 y = \frac{11 π}{6} + 2 πn

Divide both sides by

5

\frac{5 y}{5} = \frac{\frac{11 π}{6}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 y}{5} = \frac{\frac{11 π}{6}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 y}{5} : y

\frac{5 y}{5}

Divide the numbers:

\frac{5}{5} = 1

= y

Simplify

\frac{\frac{11 π}{6}}{5} + \frac{2 πn}{5} : \frac{11 π}{30} + \frac{2 πn}{5}

\frac{\frac{11 π}{6}}{5} + \frac{2 πn}{5}

\frac{\frac{11 π}{6}}{5} = \frac{11 π}{30}

\frac{\frac{11 π}{6}}{5}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 5}

Multiply the numbers:

6 \cdot 5 = 30

= \frac{11 π}{30}

= \frac{11 π}{30} + \frac{2 πn}{5}

y = \frac{11 π}{30} + \frac{2 πn}{5}

y = \frac{11 π}{30} + \frac{2 πn}{5}

y = \frac{11 π}{30} + \frac{2 πn}{5}

y = \frac{7 π}{30} + \frac{2 πn}{5}, y = \frac{11 π}{30} + \frac{2 πn}{5}

Solutions for the range

0 \leq y \leq \frac{π}{2}

y = \frac{7 π}{30}, y = \frac{11 π}{30}

Combine all the solutions

y = \frac{π}{2}, y = \frac{7 π}{30}, y = \frac{11 π}{30}

Graph

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