Homework Help! Precalculus! Three Questions!?

Question

Find all solutions of the equation in the interval [0,2pi)

2cos(x)sin(x)+sin(x)=0

sec^2(x)+tan(x)=1

cos(x)+1=sin^2(x)

Answer 1

1) 2cos(x)sin(x) + sin(x) = 0

Your first step is to factor out sin(x).

sin(x) [2cos(x) + 1] = 0

As with any equation equal to 0 with factors, equate each factor to 0 to obtain the solutions.

sin(x) = 0, which occurs at the points x = {0, pi}
2cos(x) + 1 = 0 implies
2cos(x) = -1, or
cos(x) = -1/2, which occurs at the points x = {2pi/3, 4pi/3}

Therefore, x = {0, pi, 2pi/3, 4pi/3}

2) sec^2(x) + tan(x) = 1

Your first step is to use the identity sec^2(x) = tan^2(x) + 1

tan^2(x) + 1 + tan(x) = 1

Now, subtract 1 both sides.

tan^2(x) + tan(x) = 0

Factor tan(x).

tan(x) [tan(x) + 1] = 0

tan(x) = 0, and this occurs at the points x = {0, pi}
tan(x) + 1 = 0 implies
tan(x) = -1, which occurs at the points x = {3pi/4, 7pi/4}

Therefore, our combined solutions are
x = {0, pi, 3pi/4, 7pi/4}

3) cos(x) + 1 = sin^2(x)

First, use the identity sin^2(x) = 1 - cos^2(x)

cos(x) + 1 = 1 - cos^2(x)

Subtract 1 both sides,

cos(x) = -cos^2(x)

Move -cos^2(x) to the left hand side.

cos^2(x) + cos(x) = 0

Factor.

cos(x) [cos(x) + 1] = 0, so
cos(x) = 0 (which implies x = {pi/2, 3pi/2}

cos(x) + 1 = 0,
cos(x) = -1, (which implies x = pi), so

x = {pi/2, 3pi/2, pi}

Answer 2

a million) cos²x - sin²x + sinx = 0 (a million - sin²x) - sin²x + sinx = 0 - 2sin²x + sinx + a million = 0 enable u = sinx -2u² + u + a million = 0 2u² - u - a million = 0 (2u + a million)(u - a million) = 0 u = -a million/2, a million sinx = -a million/2 and sinx = a million sparkling up for x employing arcsin. 2) The sin function is comparable to a million/2 while the argument is comparable to 30 levels, or pi/6 radians, and a hundred and fifty levels, or 5pi/6 radians: sin(2x - pi/3) = a million/2 2x - pi/3 = pi/6 2x = pi/6 + pi/3 2x = pi/6 + 2pi/6 2x = 3pi/6 2x = pi/2 x = pi/4 and sin(2x - pi/3) = a million/2 2x - pi/3 = 5pi/6 2x = 5pi/6 + pi/3 2x = 7pi/6 x = 7pi/12 strategies, x = pi/4 + 2npi, the place n is an integer and x = 7pi/12 + 2npi, the place n is an integer.

Answer 3

Use trig identities
http://math2.org/math/trig/identities.htm
to have the same function in the equation