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Popular Trigonometry >

-sqrt(2)<= sin(θ)+cos(θ)<= sqrt(2)

Solution

- 2 \leq sin (θ) + cos (θ) \leq 2

Solution

True for all θ \in R

Interval Notation

(- \infty, \infty)

Solution steps

- 2 \leq sin (θ) + cos (θ) \leq 2

a \leq u \leq b

then

a \leq u and u \leq b

- 2 \leq sin (θ) + cos (θ) and sin (θ) + cos (θ) \leq 2

- 2 \leq sin (θ) + cos (θ) :

True for all

θ \in R

- 2 \leq sin (θ) + cos (θ)

Switch sides

sin (θ) + cos (θ) \geq - 2

Use the following identity:

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

2 sin (\frac{π}{4} + θ) \geq - 2

Divide both sides by

2

2 sin (\frac{π}{4} + θ) \geq - 2

Divide both sides by

2

\frac{2 sin ( \frac{π}{4} + θ )}{2} \geq \frac{- 2}{2}

Simplify

sin (\frac{π}{4} + θ) \geq - 1

sin (\frac{π}{4} + θ) \geq - 1

Range of

sin (\frac{π}{4} + θ) : - 1 \leq sin (\frac{π}{4} + θ) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (\frac{π}{4} + θ) \leq 1

- 1 \leq sin (\frac{π}{4} + θ) \leq 1

sin (\frac{π}{4} + θ) \geq - 1 and - 1 \leq sin (\frac{π}{4} + θ) \leq 1 : - 1 \leq sin (\frac{π}{4} + θ) \leq 1

Let

y = sin (\frac{π}{4} + θ)

Combine the intervals

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

Merge Overlapping Intervals

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

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

y \geq - 1

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all θ

True for all θ \in R

sin (θ) + cos (θ) \leq 2 :

True for all

θ \in R

sin (θ) + cos (θ) \leq 2

Use the following identity:

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

2 sin (\frac{π}{4} + θ) \leq 2

Divide both sides by

2

2 sin (\frac{π}{4} + θ) \leq 2

Divide both sides by

2

\frac{2 sin ( \frac{π}{4} + θ )}{2} \leq \frac{2}{2}

Simplify

sin (\frac{π}{4} + θ) \leq 1

sin (\frac{π}{4} + θ) \leq 1

Range of

sin (\frac{π}{4} + θ) : - 1 \leq sin (\frac{π}{4} + θ) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (\frac{π}{4} + θ) \leq 1

- 1 \leq sin (\frac{π}{4} + θ) \leq 1

sin (\frac{π}{4} + θ) \leq 1 and - 1 \leq sin (\frac{π}{4} + θ) \leq 1 : - 1 \leq sin (\frac{π}{4} + θ) \leq 1

Let

y = sin (\frac{π}{4} + θ)

Combine the intervals

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

Merge Overlapping Intervals

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

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

y \leq 1

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all θ

True for all θ \in R

Combine the intervals

True for all θ \in R and True for all θ \in R

Merge Overlapping Intervals

True for all θ \in R and True for all θ \in R

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

True for all

θ \in R

and

True for all

θ \in R

True for all θ \in R

True for all θ \in R

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