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Popular Trigonometry >

sin(7)(x)<sin(2)(x)+cos(7)(x)<cos(2)(x)

Solution

sin (7) (x) < sin (2) (x) + cos (7) (x) < cos (2) (x)

Solution

No Solution

Solution steps

sin (7) (x) < sin (2) (x) + cos (7) (x) < cos (2) (x)

a < u < b

then

a < u and u < b

sin (7) (x) < sin (2) (x) + cos (7) (x) and sin (2) (x) + cos (7) (x) < cos (2) (x)

sin (7) x < sin (2) x + cos (7) x : x > 0

sin (7) x < sin (2) x + cos (7) x

Subtract

sin (2) x + cos (7) x

from both sides

sin (7) x - (sin (2) x + cos (7) x) < sin (2) x + cos (7) x - (sin (2) x + cos (7) x)

Simplify

sin (7) x - sin (2) x - cos (7) x < 0

Factor

sin (7) x - sin (2) x - cos (7) x : (sin (7) - sin (2) - cos (7)) x

sin (7) x - sin (2) x - cos (7) x

Factor out common term

x

= x (sin (7) - sin (2) - cos (7))

(sin (7) - sin (2) - cos (7)) x < 0

Multiply both sides by

- 1

(sin (7) - sin (2) - cos (7)) x < 0

Multiply both sides by -1 (reverse the inequality)

(sin (7) - sin (2) - cos (7)) x (- 1) > 0 \cdot (- 1)

Simplify

- (sin (7) - sin (2) - cos (7)) x > 0

- (sin (7) - sin (2) - cos (7)) x > 0

Divide both sides by

- sin (7) + sin (2) + cos (7)

- (sin (7) - sin (2) - cos (7)) x > 0

Divide both sides by

- sin (7) + sin (2) + cos (7)

\frac{- ( sin ( 7 ) - sin ( 2 ) - cos ( 7 ) ) x}{- sin ( 7 ) + sin ( 2 ) + cos ( 7 )} > \frac{0}{- sin ( 7 ) + sin ( 2 ) + cos ( 7 )}

Simplify

x > 0

x > 0

sin (2) x + cos (7) x < cos (2) x : x < 0

sin (2) x + cos (7) x < cos (2) x

Move

cos (2) x

to the left side

sin (2) x + cos (7) x < cos (2) x

Subtract

cos (2) x

from both sides

sin (2) x + cos (7) x - cos (2) x < cos (2) x - cos (2) x

Simplify

sin (2) x + cos (7) x - cos (2) x < 0

sin (2) x + cos (7) x - cos (2) x < 0

Factor

sin (2) x + cos (7) x - cos (2) x : (sin (2) + cos (7) - cos (2)) x

sin (2) x + cos (7) x - cos (2) x

Factor out common term

x

= x (sin (2) + cos (7) - cos (2))

(sin (2) + cos (7) - cos (2)) x < 0

Divide both sides by

sin (2) + cos (7) - cos (2)

(sin (2) + cos (7) - cos (2)) x < 0

Divide both sides by

sin (2) + cos (7) - cos (2)

\frac{( sin ( 2 ) + cos ( 7 ) - cos ( 2 ) ) x}{sin ( 2 ) + cos ( 7 ) - cos ( 2 )} < \frac{0}{sin ( 2 ) + cos ( 7 ) - cos ( 2 )}

Simplify

x < 0

x < 0

Combine the intervals

x > 0 and x < 0

Merge Overlapping Intervals

x > 0 and x < 0

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

x > 0

and

x < 0

No Solution

No Solution

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