00:01
Consider two angles, alpha and beta.
00:06
Given that tan of alpha is equal to 1 half, and cos of beta is equal to three -fifths, we want to find the exact values of a, cos of alpha plus beta, b, sign of alpha plus beta, and c tan of alpha plus beta.
00:26
To find the exact values of these expressions, we're going to use the angle sum trigonomic identities.
00:32
Which states that cos of alpha plus beta can also be written as cos of alpha time cos of beta minus sine of alpha times sine of beta similarly sign of alpha plus beta can also be written as sign of alpha times cost of beta plus course of alpha times and once we have both sine of alpha plus beta and cos of alpha plus beta, then tan of the angle sum is defined, because this is going to be equal to the sine of the angle sum, divided by the cos of the angle sum.
01:32
But right now, we only have the value of cost beta.
01:38
We need to find the values of sine beta, sine alpha, and cos alpha.
01:44
How we're going to do so is we're going to use the following tributomic identity.
01:54
Co -square of any angle, say theta, plus sine square of any angle, say theta, is equal to 1.
02:04
So we're going to use this to first find sign of beta.
02:15
How? let's rewrite this expression by solving for beta.
02:22
So we have that sine of theta is equal to 1 minus.
02:29
Close square beta to the square root.
02:35
And we have the value of cost score beta, or we have the value of cost beta, is equal to three -fifths.
02:42
So this means that sine of beta would be equal to the square root of 1 minus 9 over 25, which is equal to the square root of 16 over 25.
02:56
Simplifying to 4 fifths.
03:00
So we now have the value of sine beta.
03:03
Now let's find sine alpha and cos alpha...