What is the general solution for |cos(x)|= 1/a ?

The general solution for |cos(x)|= 1/a is No Solution for x\in\mathbb{R}

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Popular >

|cos(x)|= 1/a

Solution

∣ cos (x) ∣ = \frac{1}{a}

Solution

No Solution for x \in R

Solution steps

∣ cos (x) ∣ = \frac{1}{a}

Solve by substitution

∣ cos (x) ∣ = \frac{1}{a}

Let:

cos (x) = u

∣ u ∣ = \frac{1}{a}

∣ u ∣ = \frac{1}{a} :

No Solution for

u \in R

∣ u ∣ = \frac{1}{a}

Find positive and negative intervals

Find intervals for

∣ u ∣

u \geq 0

u \geq 0, ∣ u ∣ = u

Rewrite

∣ u ∣

for

u \geq 0 : ∣ u ∣ = u

Apply absolute rule: If

u \geq 0

then

∣ u ∣ = u

∣ u ∣ = u

u < 0

u < 0, ∣ u ∣ = - u

Rewrite

∣ u ∣

for

u < 0 : ∣ u ∣ = - u

Apply absolute rule: If

u < 0

then

∣ u ∣ = - u

∣ u ∣ = - u

Identify the intervals:

u < 0, u \geq 0

Domain of

∣ u ∣ = \frac{1}{a}

Domain definition

Find undefined (singularity) points:

True for all

u

∣ u ∣ = \frac{1}{a}

Take the denominator(s) of

∣ u ∣ = \frac{1}{a}

and compare to zero

Solve

a = 0 :

True for all

u; a = 0

a = 0

Both sides are equal

True for all u; a = 0

The following points are undefined

True for all u

Combine intervals with domain

u < 0, u \geq 0

u < 0, u \geq 0

Solve the inequality for each interval

u < 0, u \geq 0

No Solution for u \in R

Substitute back

u = cos (x)

No Solution for cos (x) \in R

No Solution for cos (x) \in R

cos (x) =

No Solution

: x = arccos (No Solution) + 2 πn, x = - arccos (No Solution) + 2 πn

cos (x) = No Solution

Apply trig inverse properties

cos (x) = No Solution

General solutions for

cos (x) =

No Solution

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

x = arccos (No Solution) + 2 πn, x = - arccos (No Solution) + 2 πn

x = arccos (No Solution) + 2 πn, x = - arccos (No Solution) + 2 πn

Combine all the solutions

x = arccos (No Solution) + 2 πn, x = - arccos (No Solution) + 2 πn

Since the equation is undefined for:

arccos (No Solution) + 2 πn, - arccos (No Solution) + 2 πn

No Solution for x \in R

Graph

Popular Examples

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sin (a) = \frac{1}{3}

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2cos(2x)-4sin(x)+1=0

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2 cos (x) - 2 sin (x) = 6

2sin^2(θ)+5sin(θ)=3

2 sin^{2} (θ) + 5 sin (θ) = 3

Frequently Asked Questions (FAQ)

What is the general solution for |cos(x)|= 1/a ?
The general solution for |cos(x)|= 1/a is No Solution for x\in\mathbb{R}