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Popular Trigonometry >

5cos(x)= 1/(cos(x))

Solution

5 cos (x) = \frac{1}{cos ( x )}

Solution

x = 1.10714 \dots + 2 πn, x = 2 π - 1.10714 \dots + 2 πn, x = 2.03444 \dots + 2 πn, x = - 2.03444 \dots + 2 πn

+1

Degrees

x = 63.43494 \dots^{\circ} + 36 0^{\circ} n, x = 296.56505 \dots^{\circ} + 36 0^{\circ} n, x = 116.56505 \dots^{\circ} + 36 0^{\circ} n, x = - 116.56505 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 cos (x) = \frac{1}{cos ( x )}

Solve by substitution

5 cos (x) = \frac{1}{cos ( x )}

Let:

cos (x) = u

5 u = \frac{1}{u}

5 u = \frac{1}{u} : u = \frac{1}{5}, u = - \frac{1}{5}

5 u = \frac{1}{u}

Multiply both sides by

u

5 u = \frac{1}{u}

Multiply both sides by

u

5 uu = \frac{1}{u} u

Simplify

5 uu : 5 u^{2}

5 uu = \frac{1}{u} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 5 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 5 u^{2}

5 u^{2} = 1

5 u^{2} = 1

Solve

5 u^{2} = 1 : u = \frac{1}{5}, u = - \frac{1}{5}

5 u^{2} = 1

Divide both sides by

5

5 u^{2} = 1

Divide both sides by

5

\frac{5 u ^{2}}{5} = \frac{1}{5}

Simplify

u^{2} = \frac{1}{5}

u^{2} = \frac{1}{5}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{5}, u = - \frac{1}{5}

u = \frac{1}{5}, u = - \frac{1}{5}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{1}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{1}{5}, u = - \frac{1}{5}

Substitute back

u = cos (x)

cos (x) = \frac{1}{5}, cos (x) = - \frac{1}{5}

cos (x) = \frac{1}{5}, cos (x) = - \frac{1}{5}

cos (x) = \frac{1}{5} : x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

cos (x) = \frac{1}{5}

Apply trig inverse properties

cos (x) = \frac{1}{5}

General solutions for

cos (x) = \frac{1}{5}

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

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

cos (x) = - \frac{1}{5} : x = arccos (- \frac{1}{5}) + 2 πn, x = - arccos (- \frac{1}{5}) + 2 πn

cos (x) = - \frac{1}{5}

Apply trig inverse properties

cos (x) = - \frac{1}{5}

General solutions for

cos (x) = - \frac{1}{5}

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

x = arccos (- \frac{1}{5}) + 2 πn, x = - arccos (- \frac{1}{5}) + 2 πn

x = arccos (- \frac{1}{5}) + 2 πn, x = - arccos (- \frac{1}{5}) + 2 πn

Combine all the solutions

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn, x = arccos (- \frac{1}{5}) + 2 πn, x = - arccos (- \frac{1}{5}) + 2 πn

Show solutions in decimal form

x = 1.10714 \dots + 2 πn, x = 2 π - 1.10714 \dots + 2 πn, x = 2.03444 \dots + 2 πn, x = - 2.03444 \dots + 2 πn

Graph

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