Question

Trigonometric Identities

50 points btw!!
Use trigonometric identities to solve each equation within the given domain.
a) –sin2(x) = cos(2x) from [–π, π]

b) 3 tan(x) = 2 sin(2x) from [0, 2π)

c) sec(x) cos(3x) = 0 from [–π/2 ,π/2]

d) 1 – cos(x) = 2 – 2 sin^2(x) from (–π, π)

e) 4cos^4 (x) – 5cos^2(x) + 1 = 0 from [0, 2π)

if you answer all thank you so much this helps a lot!

Answer 1

Explanation:

a)

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

sin(2x) = cos(2x)

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

2sin(x)cos(x) = 1 or -1

sin(x) = +/-√(1/2) or sin(x) = 0

x = (2n+1)π/4 or x = nπ

b)

3 tan(x) = 2 sin(2x)

tan(x) = 2/3 sin(2x)

sin(2x) = 3 tan(x)/2

cos^2(x) = (1 - sin^2(x)) = (1 - (3 tan(x)/2)^2)/4

x = nπ/2 + arctan(2√(2)/3) or x = nπ/2 + arctan(-2√(2)/3)

c)

sec(x) cos(3x) = 0

cos(x) = 0 or cos(3x) = 0

x = nπ/2 or x = (3n+1)π/6

d)

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

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

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

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

sin(x) = +/- √((cos(x) + 1)/2)

e)

4cos^4(x) - 5cos^2(x) + 1 = 0

cos^2(x) = (1 ± √(5))/2

x = arccos(±√((1 + √(5))/2)) or x = arccos(±√((1 - √(5))/2))

Note: n is any integer in the given domain.