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Popular >

sin(x)cos(x)>1

Solution

sin (x) cos (x) > 1

False for all x \in R

Solution steps

sin (x) cos (x) > 1

Use the following identity:

2 cos (x) sin (x) = sin (2 x)

Therefore

cos (x) sin (x) = \frac{sin ( 2 x )}{2}

\frac{sin ( 2 x )}{2} > 1

Multiply both sides by

2

\frac{sin ( 2 x )}{2} > 1

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} > 1 \cdot 2

Simplify

sin (2 x) > 2

sin (2 x) > 2

Range of

sin (2 x) : - 1 \leq sin (2 x) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (2 x) \leq 1

- 1 \leq sin (2 x) \leq 1

sin (2 x) > 2 and - 1 \leq sin (2 x) \leq 1 :

False

Let

y = sin (2 x)

Combine the intervals

y > 2 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y > 2 and - 1 \leq y \leq 1

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

y > 2

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

csc (θ) < 0

sin (x) \leq \frac{2}{2}

tan (x) - cot (x) > 0

sin^{2} (x + 3 0^{\circ}) < sin^{2} (x)

sin^{2} (x) + cos (x) \geq 1