Find all solutions to the equation in the interval [0, 2π).

cos x = sin 2x

pi divided by two., three pi divided by two.
pi divided by six., pi divided by two., five pi divided by six., three pi divided by two.
0, π
0, pi divided by six, five pi divided by six., π

x = π/6, π/2, 5π/6, 3π/2

Step-by-step explanation:

cos x = sin(2x)

Use double angle formula.

cos x = 2 sin x cos x

Move everything to one side and factor.

cos x − 2 sin x cos x = 0

cos x (1 − 2 sin x) = 0

Set each factor to 0 and solve.

cos x = 0

x = π/2, 3π/2

1 − 2 sin x = 0

sin x = 1/2

x = π/6, 5π/6

The total solution is:

x = π/6, π/2, 5π/6, 3π/2

In-center of the triangle is point N.

Step-by-step explanation:

We are given a triangle △JKL.

In triangle △JKL, they drew perpendicular bisectors of each side of the triangle.

Also we are given angle bisectors of the each of angles in given triangle.

Note: Perpendicular bisector is a line that intersect a segment into two equal parts and also perpendicular to it.

Also note that the in-center is the point forming the origin of a circle inscribed inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle.