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Popular >

sin(x)+cos(x)<= sqrt(2)

Solution

sin (x) + cos (x) \leq 2

True for all x \in R

Interval Notation

(- \infty, \infty)

Solution steps

sin (x) + cos (x) \leq 2

Use the following identity:

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

2 sin (\frac{π}{4} + x) \leq 2

Divide both sides by

2

2 sin (\frac{π}{4} + x) \leq 2

Divide both sides by

2

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

Simplify

sin (\frac{π}{4} + x) \leq 1

sin (\frac{π}{4} + x) \leq 1

Range of

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

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (\frac{π}{4} + x) \leq 1

- 1 \leq sin (\frac{π}{4} + x) \leq 1

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

Let

y = sin (\frac{π}{4} + x)

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 x

True for all x \in R

3 tan^{2} (x) + 3 tan (x) > 0

- 0.25 \leq 0.5 sin (2 x)

2 sin (x) - 3 \geq 0

4 - tan (\frac{θ}{2}) > 3

sin (x) > - \frac{2}{2}