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Popular Trigonometry >

solvefor t,cos(wt+k)=(20)/((2pif)^2)

Solution

solvefor

Solution

t = \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}, t = - \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}

+1

Radians

t = \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} - \frac{k}{w} + \frac{2 π}{w} n, t = - \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} - \frac{k}{w} + \frac{2 π}{w} n

Solution steps

cos (wt + k) = \frac{20}{( 2 π f ) ^{2}}

Simplify

\frac{20}{( 2 π f ) ^{2}} : \frac{5}{π ^{2} f ^{2}}

\frac{20}{( 2 π f ) ^{2}}

(2 π f)^{2} = 2^{2} π^{2} f^{2}

(2 π f)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} π^{2} f^{2}

= \frac{20}{2 ^{2} π ^{2} f ^{2}}

Factor

20 : 2^{2} \cdot 5

Factor

20 = 2^{2} \cdot 5

= \frac{2 ^{2} \cdot 5}{2 ^{2} π ^{2} f ^{2}}

Cancel the common factor:

2^{2}

= \frac{5}{π ^{2} f ^{2}}

Apply trig inverse properties

cos (wt + k) = \frac{5}{π ^{2} f ^{2}}

General solutions for

cos (wt + k) = \frac{5}{π ^{2} f ^{2}}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = - arccos (a) + 2 πn

wt + k = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn, wt + k = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn

wt + k = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn, wt + k = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn

Solve

wt + k = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn : t = \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}; w \neq = 0

wt + k = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn

Move

k

to the right side

wt + k = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn

Subtract

k

from both sides

wt + k - k = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn - k

Simplify

wt = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn - k

wt = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn - k

Divide both sides by

w; w \neq = 0

wt = arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn - k

Divide both sides by

w; w \neq = 0

\frac{wt}{w} = \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}; w \neq = 0

Simplify

t = \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}; w \neq = 0

t = \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}; w \neq = 0

Solve

wt + k = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn : t = - \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}; w \neq = 0

wt + k = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn

Move

k

to the right side

wt + k = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn

Subtract

k

from both sides

wt + k - k = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn - k

Simplify

wt = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn - k

wt = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn - k

Divide both sides by

w; w \neq = 0

wt = - arccos (\frac{5}{π ^{2} f ^{2}}) + 2 πn - k

Divide both sides by

w; w \neq = 0

\frac{wt}{w} = - \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}; w \neq = 0

Simplify

t = - \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}; w \neq = 0

t = - \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}; w \neq = 0

t = \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}, t = - \frac{arccos ( \frac{5}{π ^{2} f ^{2}} )}{w} + \frac{2 πn}{w} - \frac{k}{w}

Graph

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