Double angle formula question

Question

Use the double angle formulas to find sin 2x, cos 2x, and tan 2x. cot x= 2/3, sin x > 0

Answer 1

sin2x = 12/13 = 0.923

cos2x =-5/13 =-0.385

tan2x =-2.40

sin2x = 2sinxcosx

cos2x = cos^2x - sin^2x

tan2x = sin2x/cos2x

cotx = 2/3 = adjacent side/opposite side.

hypotenuse = h = sqr(2^2+ 3^2) = sqr(4+9)= sqr13

sinx= 3/sqr13

cosx = 2/sqr13

plug those values into the double angle formulas.

to check your answer calculate x. If cotx =2/3, tanx = 3/2 =1.5

and x= 56.31 degrees. 2x = 112.62 degrees,

use a calculator with trig functions to get:

sin112.62 = 0.923 = 12/13

cos112.62= -0.385 = -5/13

tan112.62 = -2.4 = 12/-5

Answer 2

Given:

cot x = 2/3, sin x > 0

If sin x > 0, our angle in a coordinate plane is either in first or second quadrant.

If cot x =2/3, our angle in a coordinate plane is either in first or third quadrant.

Therefore, the angle that we are looking for is definitely in the first quadrant (where both x and y are greater than 0)

cot x = adjacent/ opposite = x/y = 2/3

it means

x=2

y=3

To find r, use Pythagorean Theorem:

r = (x² + y²)^1/2

r = (2² + 3²)^1/2 = (13)^1/2

Therefore:

sin x = 3/√13

cos x = 2/√13

tan x = 3/2

The double angle identities are:

sin 2x = 2•cos x• sin x

cos 2x = 1-2 sin²x

tan 2x = 2•tan x/(1-tan² x)

Answers:

sin 2x = 2(2/√13)(3/√13) =12/13

cos 2x = 1-2(3/√13)² = 1-2(9/13)=1-18/13 = -5/13

tan 2x= 2(3/2)/(1-(3/2)²) =3/(1-9/4) = 3/(-5/4) = -12/5

Answer 3

Draw and label a diagram!

Use Pythagoras to determine sin x and cos x.

Substitute values into appropriate formula.