Sin theta equals 5/12 in quadrant, find the exact value of cos 2theta.
I'm not sure how to solve this question, I need some help.
From sinØ = 5/12
make a sketch and use Pythagoras to find the missing side x
x^2 + 5^2 = 12^2
cos 2Ø = 1 - 2 sin^2 Ø
= 1 - 2(5/12)^2
= 1 - 50/144
= 94/144
= 47/72
Did not even have to find the third side, just followed my usual routine.
To find the exact value of cos 2theta, we first need to find the value of theta. We know that sin theta equals 5/12 and that theta is in a specific quadrant, but you haven't specified the quadrant. The value of theta will determine the value of cos 2theta.
If the value of sin theta is positive (5/12 is positive) and the quadrant is between 0 and 90 degrees, we can solve for theta as follows:
sin theta = 5/12
Using the Pythagorean Identity (sin^2 theta + cos^2 theta = 1):
(5/12)^2 + cos^2 theta = 1
25/144 + cos^2 theta = 1
cos^2 theta = 1 - 25/144
cos^2 theta = 144/144 - 25/144
cos^2 theta = 119/144
Taking the square root of both sides:
cos theta = ±√(119/144)
Since the value of sin is positive, we can determine that cos theta is positive as well. Therefore:
cos theta = √(119/144)
Now that we have the value of cos theta, we can find cos 2theta using the double angle formula for cosine:
cos 2theta = cos^2 theta - sin^2 theta
Substituting the values we found earlier:
cos 2theta = (√(119/144))^2 - (5/12)^2
cos 2theta = (119/144) - (25/144)
cos 2theta = (119 - 25)/144
cos 2theta = 94/144
Simplifying further:
cos 2theta = 47/72
So, the exact value of cos 2theta is 47/72.