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Popular >

cos(x^4)+sin(x^4)<= 0.5

Solution

cos (x^{4}) + sin (x^{4}) \leq 0.5

False for all x \in R

Solution steps

cos (x^{4}) + sin (x^{4}) \leq 0.5

Use the following identity:

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

2 sin (\frac{π}{4} + x^{4}) \leq 0.5

Divide both sides by

2

2 sin (\frac{π}{4} + x^{4}) \leq 0.5

Divide both sides by

2

\frac{2 sin ( \frac{π}{4} + x ^{4} )}{2} \leq \frac{0.5}{2}

Simplify

\frac{2 sin ( \frac{π}{4} + x ^{4} )}{2} \leq \frac{0.5}{2}

Simplify

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

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

Cancel the common factor:

2

= sin (\frac{π}{4} + x^{4})

Simplify

\frac{0.5}{2} : 0.35355 \dots

\frac{0.5}{2}

Convert element to a decimal form

2 = 1.41421 \dots

= \frac{0.5}{1.41421 \dots}

Divide the numbers:

\frac{0.5}{1.41421 \dots} = 0.35355 \dots

= 0.35355 \dots

sin (\frac{π}{4} + x^{4}) \leq 0.35355 \dots

sin (\frac{π}{4} + x^{4}) \leq 0.35355 \dots

sin (\frac{π}{4} + x^{4}) \leq 0.35355 \dots

Range of

sin (\frac{π}{4} + x^{4}) :

False for all

y \in R

sin (\frac{π}{4} + x^{4}) \leq 0.35355 \dots and

False for all

y \in R :

False

Let

y = sin (\frac{π}{4} + x^{4})

Combine the intervals

y \leq 0.35355 \dots and False for all y \in R

Merge Overlapping Intervals

y \leq 0.35355 \dots and False for all y \in R

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

y \leq 0.35355 \dots

and

False for all

y \in R

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

sin (3 X - 15) > - \frac{2}{2}

2 (tan (x) + 1) + 3 (1 - tan (x)) < - 2 (tan (x) - 3)

cot (x) > 1

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

sin (x) \cdot cos (2 x) > 0