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Popular >

0=1.9cos(0.1t)-1.25sin(0.2t)

Solution

0 = 1.9 cos (0.1 t) - 1.25 sin (0.2 t)

Solution

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{0.86331 \dots + 2 πn}{0.1}, t = \frac{π - 0.86331 \dots + 2 πn}{0.1}

Degrees

t = 90 0^{\circ} + 360 0^{\circ} n, t = 270 0^{\circ} + 360 0^{\circ} n, t = 494.64197 \dots^{\circ} + 360 0^{\circ} n, t = 1305.35802 \dots^{\circ} + 360 0^{\circ} n

Solution steps

0 = 1.9 cos (0.1 t) - 1.25 sin (0.2 t)

Switch sides

1.9 cos (0.1 t) - 1.25 sin (0.2 t) = 0

Let:

u = 0.1 t

1.9 cos (u) - 1.25 sin (2 u) = 0

Rewrite using trig identities

- 1.25 sin (2 u) + 1.9 cos (u)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= - 1.25 \cdot 2 sin (u) cos (u) + 1.9 cos (u)

Simplify

= - 2.5 sin (u) cos (u) + 1.9 cos (u)

1.9 cos (u) - 2.5 cos (u) sin (u) = 0

Factor

1.9 cos (u) - 2.5 cos (u) sin (u) : - cos (u) (2.5 sin (u) - 1.9)

1.9 cos (u) - 2.5 cos (u) sin (u)

Factor out common term

- cos (u)

= - cos (u) (- 1.9 + 2.5 sin (u))

- cos (u) (2.5 sin (u) - 1.9) = 0

Solving each part separately

cos (u) = 0 or 2.5 sin (u) - 1.9 = 0

cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = 0

General solutions for

cos (u) = 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}

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

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

2.5 sin (u) - 1.9 = 0 : u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

2.5 sin (u) - 1.9 = 0

Multiply both sides by

10

2.5 sin (u) - 1.9 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

2.5 sin (u) \cdot 10 - 1.9 \cdot 10 = 0 \cdot 10

Refine

25 sin (u) - 19 = 0

25 sin (u) - 19 = 0

Move

19

to the right side

25 sin (u) - 19 = 0

Add

19

to both sides

25 sin (u) - 19 + 19 = 0 + 19

Simplify

25 sin (u) = 19

25 sin (u) = 19

Divide both sides by

25

25 sin (u) = 19

Divide both sides by

25

\frac{25 sin ( u )}{25} = \frac{19}{25}

Simplify

sin (u) = \frac{19}{25}

sin (u) = \frac{19}{25}

Apply trig inverse properties

sin (u) = \frac{19}{25}

General solutions for

sin (u) = \frac{19}{25}

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

u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

Combine all the solutions

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

Substitute back

u = 0.1 t

0.1 t = \frac{π}{2} + 2 πn : t = 5 π + 20 πn

0.1 t = \frac{π}{2} + 2 πn

Multiply both sides by

10

0.1 t = \frac{π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.1 t \cdot 10 = \frac{π}{2} \cdot 10 + 2 πn \cdot 10

Refine

t = 5 π + 20 πn

t = 5 π + 20 πn

0.1 t = \frac{3 π}{2} + 2 πn : t = 15 π + 20 πn

0.1 t = \frac{3 π}{2} + 2 πn

Multiply both sides by

10

0.1 t = \frac{3 π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.1 t \cdot 10 = \frac{3 π}{2} \cdot 10 + 2 πn \cdot 10

Refine

t = 15 π + 20 πn

t = 15 π + 20 πn

0.1 t = arcsin (\frac{19}{25}) + 2 πn : t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = arcsin (\frac{19}{25}) + 2 πn

Divide both sides by

0.1

0.1 t = arcsin (\frac{19}{25}) + 2 πn

Divide both sides by

0.1

\frac{0.1 t}{0.1} = \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Simplify

\frac{0.1 t}{0.1} = \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Simplify

\frac{0.1 t}{0.1} : t

\frac{0.1 t}{0.1}

Cancel the common factor:

0.1

= t

Simplify

\frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1} : \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

\frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Apply rule

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

= \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn : t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn

Divide both sides by

0.1

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn

Divide both sides by

0.1

\frac{0.1 t}{0.1} = \frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Simplify

\frac{0.1 t}{0.1} = \frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Simplify

\frac{0.1 t}{0.1} : t

\frac{0.1 t}{0.1}

Cancel the common factor:

0.1

= t

Simplify

\frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1} : \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

\frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

Apply rule

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

= \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}, t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

Show solutions in decimal form

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{0.86331 \dots + 2 πn}{0.1}, t = \frac{π - 0.86331 \dots + 2 πn}{0.1}

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