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Popular >

-1<= (pi(cos(x)-sin(x)))/(4sqrt(2))<= 1

Solution

- 1 \leq \frac{π ( cos ( x ) - sin ( x ) )}{4 2} \leq 1

Solution

True for all x \in R

Interval Notation

(- \infty, \infty)

Solution steps

- 1 \leq \frac{π ( cos ( x ) - sin ( x ) )}{4 2} \leq 1

a \leq u \leq b

then

a \leq u and u \leq b

- 1 \leq \frac{π ( cos ( x ) - sin ( x ) )}{4 2} and \frac{π ( cos ( x ) - sin ( x ) )}{4 2} \leq 1

- 1 \leq \frac{π ( cos ( x ) - sin ( x ) )}{4 2} :

True for all

x \in R

- 1 \leq \frac{π ( cos ( x ) - sin ( x ) )}{4 2}

Switch sides

\frac{π ( cos ( x ) - sin ( x ) )}{4 2} \geq - 1

Use the following identity:

cos (x) - sin (x) = 2 cos (\frac{π}{4} + x)

\frac{π 2 cos ( \frac{π}{4} + x )}{4 2} \geq - 1

Simplify

\frac{π 2 cos ( \frac{π}{4} + x )}{4 2} : \frac{π}{4} cos (x + \frac{π}{4})

\frac{π 2 cos ( \frac{π}{4} + x )}{4 2}

Cancel the common factor:

2

= \frac{π cos ( x + \frac{π}{4} )}{4}

= \frac{π}{4} cos (x + \frac{π}{4})

\frac{π}{4} cos (x + \frac{π}{4}) \geq - 1

Multiply both sides by

4

\frac{π}{4} cos (x + \frac{π}{4}) \geq - 1

Multiply both sides by

4

4 \cdot \frac{π}{4} cos (x + \frac{π}{4}) \geq 4 (- 1)

Simplify

π cos (x + \frac{π}{4}) \geq - 4

π cos (x + \frac{π}{4}) \geq - 4

Divide both sides by

π

π cos (x + \frac{π}{4}) \geq - 4

Divide both sides by

π

\frac{π cos ( x + \frac{π}{4} )}{π} \geq \frac{- 4}{π}

Simplify

cos (x + \frac{π}{4}) \geq - \frac{4}{π}

cos (x + \frac{π}{4}) \geq - \frac{4}{π}

Range of

cos (x + \frac{π}{4}) : - 1 \leq cos (x + \frac{π}{4}) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x + \frac{π}{4}) \leq 1

- 1 \leq cos (x + \frac{π}{4}) \leq 1

cos (x + \frac{π}{4}) \geq - \frac{4}{π} and - 1 \leq cos (x + \frac{π}{4}) \leq 1 : - 1 \leq cos (x + \frac{π}{4}) \leq 1

Let

y = cos (x + \frac{π}{4})

Combine the intervals

y \geq - \frac{4}{π} and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \geq - \frac{4}{π} and - 1 \leq y \leq 1

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

y \geq - \frac{4}{π}

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

\frac{π ( cos ( x ) - sin ( x ) )}{4 2} \leq 1 :

True for all

x \in R

\frac{π ( cos ( x ) - sin ( x ) )}{4 2} \leq 1

Use the following identity:

cos (x) - sin (x) = 2 cos (\frac{π}{4} + x)

\frac{π 2 cos ( \frac{π}{4} + x )}{4 2} \leq 1

Simplify

\frac{π 2 cos ( \frac{π}{4} + x )}{4 2} : \frac{π}{4} cos (x + \frac{π}{4})

\frac{π 2 cos ( \frac{π}{4} + x )}{4 2}

Cancel the common factor:

2

= \frac{π cos ( x + \frac{π}{4} )}{4}

= \frac{π}{4} cos (x + \frac{π}{4})

\frac{π}{4} cos (x + \frac{π}{4}) \leq 1

Multiply both sides by

4

\frac{π}{4} cos (x + \frac{π}{4}) \leq 1

Multiply both sides by

4

4 \cdot \frac{π}{4} cos (x + \frac{π}{4}) \leq 1 \cdot 4

Simplify

π cos (x + \frac{π}{4}) \leq 4

π cos (x + \frac{π}{4}) \leq 4

Divide both sides by

π

π cos (x + \frac{π}{4}) \leq 4

Divide both sides by

π

\frac{π cos ( x + \frac{π}{4} )}{π} \leq \frac{4}{π}

Simplify

cos (x + \frac{π}{4}) \leq \frac{4}{π}

cos (x + \frac{π}{4}) \leq \frac{4}{π}

Range of

cos (x + \frac{π}{4}) : - 1 \leq cos (x + \frac{π}{4}) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x + \frac{π}{4}) \leq 1

- 1 \leq cos (x + \frac{π}{4}) \leq 1

cos (x + \frac{π}{4}) \leq \frac{4}{π} and - 1 \leq cos (x + \frac{π}{4}) \leq 1 : - 1 \leq cos (x + \frac{π}{4}) \leq 1

Let

y = cos (x + \frac{π}{4})

Combine the intervals

y \leq \frac{4}{π} and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \leq \frac{4}{π} and - 1 \leq y \leq 1

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

y \leq \frac{4}{π}

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 \in R

Popular Examples

0<= cos(θ)<= 1

0 \leq cos (θ) \leq 1

-2sin^2(x)0<x<360

- 2 sin^{2} (x) 0 < x < 360

-1<= tan(x/2-pi/3)<= sqrt(3)

- 1 \leq tan (\frac{x}{2} - \frac{π}{3}) \leq 3

-1>=-cos(2x)>= 1

- 1 \geq - cos (2 x) \geq 1

0<82.5-67.5cos(pi/6 t)<20

0 < 82.5 - 67.5 cos (\frac{π}{6} t) < 20