Given sin x=5/13 and 0<= x <= π/2 find:
a) sin 2x
b) cos 2x
c) tan 2x
you have a 5-12-13 triangle, so
cosx = 12/13
tanx = 5/12
Now just use your double angle formulas. For instance,
tan 2x = 2tanx/(1-tan^2 x) = 2(5/12)/(1 - (5/12)^2) = 120/119
To find the values of sin 2x, cos 2x, and tan 2x, we need to use the double-angle identities. These identities allow us to express trigonometric functions in terms of other trigonometric functions.
Let's start with calculating sin 2x:
a) sin 2x = 2sin x * cos x
To find sin x in the given range of 0 <= x <= π/2, we use the fact that sin x = 5/13.
sin 2x = 2 * (5/13) * cos x
Now, we need to find cos x. To do that, we can use the Pythagorean identity: sin^2x + cos^2x = 1.
Since sin x = 5/13, we can substitute this value into the equation:
(5/13)^2 + cos^2x = 1
Simplifying the equation gives us:
25/169 + cos^2x = 1
cos^2x = 144/169
cos x = ±12/13
However, since 0 <= x <= π/2, we know that cos x must be positive. So, cos x = 12/13.
We can now substitute the values of sin x and cos x into the equation for sin 2x:
sin 2x = 2 * (5/13) * (12/13)
sin 2x = 120/169
Therefore, the value of sin 2x is 120/169.
b) cos 2x = cos^2x - sin^2x
We already found the value of cos x to be 12/13. Let's substitute that and the value of sin x into the equation:
cos 2x = (12/13)^2 - (5/13)^2
Simplifying gives us:
cos 2x = 144/169 - 25/169
cos 2x = 119/169
Therefore, the value of cos 2x is 119/169.
c) tan 2x = sin 2x / cos 2x
We can substitute the values we found for sin 2x and cos 2x into the equation:
tan 2x = (120/169) / (119/169)
tan 2x = 120/119
Therefore, the value of tan 2x is 120/119.