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Popular Trigonometry >

0=cos(20t)-cos(20.02t)

Solution

0 = cos (20 t) - cos (20.02 t)

Solution

t = 200 πn, t = 100 π + 200 πn, t = \frac{200 πn}{2001}, t = \frac{100 π}{2001} + \frac{200 πn}{2001}

Degrees

t = 0^{\circ} + 3600 0^{\circ} n, t = 1800 0^{\circ} + 3600 0^{\circ} n, t = 0^{\circ} + 17.99100 \dots^{\circ} n, t = 8.99550 \dots^{\circ} + 17.99100 \dots^{\circ} n

Solution steps

0 = cos (20 t) - cos (20.02 t)

Switch sides

cos (20 t) - cos (20.02 t) = 0

Rewrite using trig identities

- cos (20.02 t) + cos (20 t)

Use the Sum to Product identity:

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

= - 2 sin (\frac{20 t + 20.02 t}{2}) sin (\frac{20 t - 20.02 t}{2})

Simplify

- 2 sin (\frac{20 t + 20.02 t}{2}) sin (\frac{20 t - 20.02 t}{2}) : 2 sin (0.01 t) sin (20.01 t)

- 2 sin (\frac{20 t + 20.02 t}{2}) sin (\frac{20 t - 20.02 t}{2})

Add similar elements:

20 t + 20.02 t = 40.02 t

= - 2 sin (\frac{40.02 t}{2}) sin (\frac{20 t - 20.02 t}{2})

\frac{20 t - 20.02 t}{2} = - \frac{0.02 t}{2}

\frac{20 t - 20.02 t}{2}

Add similar elements:

20 t - 20.02 t = - 0.02 t

= \frac{- 0.02 t}{2}

Apply the fraction rule:

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

= - \frac{0.02 t}{2}

= - 2 sin (\frac{40.02 t}{2}) sin (- \frac{0.02 t}{2})

Use the negative angle identity:

sin (- x) = - sin (x)

= - 2 (- sin (\frac{0.02 t}{2})) sin (\frac{40.02 t}{2})

Apply rule

- (- a) = a

= 2 sin (\frac{0.02 t}{2}) sin (\frac{40.02 t}{2})

Divide the numbers:

\frac{0.02}{2} = 0.01

= 2 sin (0.01 t) sin (\frac{40.02 t}{2})

Divide the numbers:

\frac{40.02}{2} = 20.01

= 2 sin (0.01 t) sin (20.01 t)

= 2 sin (0.01 t) sin (20.01 t)

2 sin (0.01 t) sin (20.01 t) = 0

Solving each part separately

sin (0.01 t) = 0 or sin (20.01 t) = 0

sin (0.01 t) = 0 : t = 200 πn, t = 100 π + 200 πn

sin (0.01 t) = 0

General solutions for

sin (0.01 t) = 0

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}

0.01 t = 0 + 2 πn, 0.01 t = π + 2 πn

0.01 t = 0 + 2 πn, 0.01 t = π + 2 πn

Solve

0.01 t = 0 + 2 πn : t = 200 πn

0.01 t = 0 + 2 πn

0 + 2 πn = 2 πn

0.01 t = 2 πn

Multiply both sides by

100

0.01 t = 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.01 t \cdot 100 = 2 πn \cdot 100

Refine

t = 200 πn

t = 200 πn

Solve

0.01 t = π + 2 πn : t = 100 π + 200 πn

0.01 t = π + 2 πn

Multiply both sides by

100

0.01 t = π + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.01 t \cdot 100 = π 100 + 2 πn \cdot 100

Refine

t = 100 π + 200 πn

t = 100 π + 200 πn

t = 200 πn, t = 100 π + 200 πn

sin (20.01 t) = 0 : t = \frac{200 πn}{2001}, t = \frac{100 π}{2001} + \frac{200 πn}{2001}

sin (20.01 t) = 0

General solutions for

sin (20.01 t) = 0

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}

20.01 t = 0 + 2 πn, 20.01 t = π + 2 πn

20.01 t = 0 + 2 πn, 20.01 t = π + 2 πn

Solve

20.01 t = 0 + 2 πn : t = \frac{200 πn}{2001}

20.01 t = 0 + 2 πn

0 + 2 πn = 2 πn

20.01 t = 2 πn

Multiply both sides by

100

20.01 t = 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

20.01 t \cdot 100 = 2 πn \cdot 100

Refine

2001 t = 200 πn

2001 t = 200 πn

Divide both sides by

2001

2001 t = 200 πn

Divide both sides by

2001

\frac{2001 t}{2001} = \frac{200 πn}{2001}

Simplify

t = \frac{200 πn}{2001}

t = \frac{200 πn}{2001}

Solve

20.01 t = π + 2 πn : t = \frac{100 π}{2001} + \frac{200 πn}{2001}

20.01 t = π + 2 πn

Multiply both sides by

100

20.01 t = π + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

20.01 t \cdot 100 = π 100 + 2 πn \cdot 100

Refine

2001 t = 100 π + 200 πn

2001 t = 100 π + 200 πn

Divide both sides by

2001

2001 t = 100 π + 200 πn

Divide both sides by

2001

\frac{2001 t}{2001} = \frac{100 π}{2001} + \frac{200 πn}{2001}

Simplify

t = \frac{100 π}{2001} + \frac{200 πn}{2001}

t = \frac{100 π}{2001} + \frac{200 πn}{2001}

t = \frac{200 πn}{2001}, t = \frac{100 π}{2001} + \frac{200 πn}{2001}

Combine all the solutions

t = 200 πn, t = 100 π + 200 πn, t = \frac{200 πn}{2001}, t = \frac{100 π}{2001} + \frac{200 πn}{2001}

Graph

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