find sin 2x
if tan x=2 cosx>0
first quadrant because sin and cos are +
triangle is 1, 2, sqrt 45
sin x = 2/sqrt 5
cos x = 1/sqrt 5
I think you can take it from there
since both the cosine and the tangent are positive, x must be an angle in the first quadrant.
from tanx = 2/1, draw a right-angled triangle with angle x at the origin that has a height of 2, and a base of 1
(tangentx = opposite/adjacent, so the opposite is 2 and the adjacent is 1)
so by Pythagoras, the hypotenuse is √5
so sinx - 2/√5 and cosx = 1/√5
sin2x = 2sinxcosx (one of our general identities)
= 2(2/√5)(1/√5) = 4/5
you can check this on the calculator
take inverse tan of 2 to get the angle,
double that angle
take the sine
you should get .8 which is 4/5
To find the value of sin 2x, we can use the trigonometric identity for sin 2x:
sin 2x = 2 sin x cos x
Given that tan x = 2 and cos x > 0, we can determine the values of sin x and cos x using the given information.
Since tan x = sin x / cos x, we can rearrange this as sin x = tan x * cos x.
Substituting the given values, we have sin x = 2 * cos x.
Next, let's find cos x. Since cos x > 0, we can determine the value of cos x based on the quadrant of the angle x.
Since tan x = 2 and tan x = sin x / cos x, we can substitute the given value for tan x and solve for cos x:
2 = sin x / cos x
Multiplying both sides by cos x:
2 * cos x = sin x
Since sin^2 x + cos^2 x = 1 (this is a Pythagorean identity), we can substitute the values we just found:
(2 * cos x)^2 + cos^2 x = 1
Simplifying:
4 * cos^2 x + cos^2 x = 1
5 * cos^2 x = 1
Dividing by 5:
cos^2 x = 1/5
Taking the square root of both sides (remembering that cos x is positive):
cos x = sqrt(1/5)
Now that we have the value of cos x, we can substitute it back into the equation for sin x:
sin x = 2 * cos x = 2 * sqrt(1/5) = 2 * sqrt(5) / 5
Finally, substitute the values of sin x and cos x into the sin 2x formula:
sin 2x = 2 * sin x * cos x = 2 * (2 * sqrt(5) / 5) * sqrt(1/5) = 4 * sqrt(5) / 5