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Popular Trigonometry >

cos(x)+sin(x)<0.46

Solution

cos (x) + sin (x) < 0.46

Solution

- 4.25828 \dots + 6.28318 \dots n < x < 0.33129 \dots - \frac{π}{4} + 6.28318 \dots n

Interval Notation

(- 4.25828 \dots + 6.28318 \dots n, 0.33129 \dots - \frac{π}{4} + 6.28318 \dots n)

Decimal

- 4.25828 \dots + 6.28318 \dots n < x < - 0.45410 \dots + 6.28318 \dots n

Solution steps

cos (x) + sin (x) < 0.46

Use the following identity:

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

2 sin (\frac{π}{4} + x) < 0.46

Divide both sides by

2

2 sin (\frac{π}{4} + x) < 0.46

Divide both sides by

2

\frac{2 sin ( \frac{π}{4} + x )}{2} < \frac{0.46}{2}

Simplify

\frac{2 sin ( \frac{π}{4} + x )}{2} < \frac{0.46}{2}

Simplify

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

\frac{2 sin ( \frac{π}{4} + x )}{2}

Cancel the common factor:

2

= sin (\frac{π}{4} + x)

Simplify

\frac{0.46}{2} : 0.32526 \dots

\frac{0.46}{2}

Convert element to a decimal form

2 = 1.41421 \dots

= \frac{0.46}{1.41421 \dots}

Divide the numbers:

\frac{0.46}{1.41421 \dots} = 0.32526 \dots

= 0.32526 \dots

sin (\frac{π}{4} + x) < 0.32526 \dots

sin (\frac{π}{4} + x) < 0.32526 \dots

sin (\frac{π}{4} + x) < 0.32526 \dots

For

sin (x) < a

, if

- 1 < a \leq 1

then

- π - arcsin (a) + 2 πn < x < arcsin (a) + 2 πn

- π - arcsin (0.32526 \dots) + 2 πn < (\frac{π}{4} + x) < arcsin (0.32526 \dots) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (0.32526 \dots) + 2 πn < \frac{π}{4} + x and \frac{π}{4} + x < arcsin (0.32526 \dots) + 2 πn

- π - arcsin (0.32526 \dots) + 2 πn < \frac{π}{4} + x : x > - 0.33129 \dots - \frac{5 π}{4} + 6.28318 \dots n

- π - arcsin (0.32526 \dots) + 2 πn < \frac{π}{4} + x

Switch sides

\frac{π}{4} + x > - π - arcsin (0.32526 \dots) + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} + x > - π - arcsin (0.32526 \dots) + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} + x - \frac{π}{4} > - π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Simplify

x > - π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

x > - π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Simplify

x > - π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Simplify

- π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4} : - π - 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

- π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

arcsin (0.32526 \dots) = 0.33129 \dots

= - π - 0.33129 \dots + 2 πn - \frac{π}{4}

Multiply the numbers:

2 \cdot 3.14159 \dots = 6.28318 \dots

= - π - 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

x > - π - 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

x > - π - 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

Simplify

- π - 0.33129 \dots - \frac{π}{4} : - 0.33129 \dots - \frac{5 π}{4}

- π - 0.33129 \dots - \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= - \frac{π 4}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- π 4 - π}{4}

Add similar elements:

- 4 π - π = - 5 π

= \frac{- 5 π}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - 0.33129 \dots - \frac{5 π}{4}

x > - 0.33129 \dots - \frac{5 π}{4} + 6.28318 \dots n

\frac{π}{4} + x < arcsin (0.32526 \dots) + 2 πn : x < 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

\frac{π}{4} + x < arcsin (0.32526 \dots) + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} + x < arcsin (0.32526 \dots) + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} + x - \frac{π}{4} < arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Simplify

x < arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

x < arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Simplify

x < arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Simplify

arcsin (0.32526 \dots) + 2 πn - \frac{π}{4} : 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

arcsin (0.32526 \dots) = 0.33129 \dots

= 0.33129 \dots + 2 πn - \frac{π}{4}

Multiply the numbers:

2 \cdot 3.14159 \dots = 6.28318 \dots

= 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

x < 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

x < 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

Combine the intervals

x > - 0.33129 \dots - \frac{5 π}{4} + 6.28318 \dots n and x < 0.33129 \dots + 6.28318 \dots n - \frac{π}{4}

Merge Overlapping Intervals

- 4.25828 \dots + 6.28318 \dots n < x < 0.33129 \dots - \frac{π}{4} + 6.28318 \dots n

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