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Popular Trigonometry >

(cos(x)-1.5708)(cos(x)+1.5708)<= 0

Solution

(cos (x) - 1.5708) (cos (x) + 1.5708) \leq 0

Solution

True for all x

Interval Notation

(- \infty, \infty)

Solution steps

(cos (x) - 1.5708) (cos (x) + 1.5708) \leq 0

Let:

u = cos (x)

(u - 1.5708) (u + 1.5708) \leq 0

(u - 1.5708) (u + 1.5708) \leq 0 : - 1.5708 \leq u \leq 1.5708

(u - 1.5708) (u + 1.5708) \leq 0

Expand

(u - 1.5708) (u + 1.5708) : u^{2} - 2.46741264

(u - 1.5708) (u + 1.5708)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = u, b = 1.5708

= u^{2} - 1.570 8^{2}

1.570 8^{2} = 2.46741264

= u^{2} - 2.46741264

u^{2} - 2.46741264 \leq 0

Move

2.46741264

to the right side

u^{2} - 2.46741264 \leq 0

Add

2.46741264

to both sides

u^{2} - 2.46741264 + 2.46741264 \leq 0 + 2.46741264

Simplify

u^{2} \leq 2.46741264

u^{2} \leq 2.46741264

For

u^{n} \leq a

, if

n

is even then

- 2.46741264 \leq u \leq 2.46741264

2.46741264 = 1.5708

2.46741264

2.46741264 = 1.5708

= 1.5708

- 1.5708 \leq u \leq 1.5708

- 1.5708 \leq u \leq 1.5708

Substitute back

u = cos (x)

- 1.5708 \leq cos (x) \leq 1.5708

a \leq u \leq b

then

a \leq u and u \leq b

- 1.5708 \leq cos (x) and cos (x) \leq 1.5708

- 1.5708 \leq cos (x) :

True for all

x \in R

- 1.5708 \leq cos (x)

Switch sides

cos (x) \geq - 1.5708

Range of

cos (x) : - 1 \leq cos (x) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x) \leq 1

- 1 \leq cos (x) \leq 1

cos (x) \geq - 1.5708 and - 1 \leq cos (x) \leq 1 : - 1 \leq cos (x) \leq 1

Let

y = cos (x)

Combine the intervals

y \geq - 1.5708 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \geq - 1.5708 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y \geq - 1.5708

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

cos (x) \leq 1.5708 :

True for all

x \in R

cos (x) \leq 1.5708

Range of

cos (x) : - 1 \leq cos (x) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x) \leq 1

- 1 \leq cos (x) \leq 1

cos (x) \leq 1.5708 and - 1 \leq cos (x) \leq 1 : - 1 \leq cos (x) \leq 1

Let

y = cos (x)

Combine the intervals

y \leq 1.5708 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \leq 1.5708 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y \leq 1.5708

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

Combine the intervals

True for all x \in R and True for all x \in R

Merge Overlapping Intervals

True for all x \in R and True for all x \in R

The intersection of two intervals is the set of numbers which are in both intervals

True for all

x \in R

and

True for all

x \in R

True for all x \in R

True for all x

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