Half angle identities
Find exact values of
a) cos 1/2
b) sin 1/2
c) tan 1/2
based on:
cos = 1/5
greater than 270 deg less than 360 deg.
I get a) - rt 3/5 b) rt24/5 and c) does not work out at all ....
Help please..
book says for answers
a)-rt15/5 b) rt10/5 c) -rt6/3
How do you get that answers ?
It's hard to interpret your expressions, with no parentheses.
Using θ as the reference angle with cosθ = 1/5, your angle Ø in the 4th quadrant is Ø = 2pi - θ
For (a) if you got -√(3/5), that is the same as -√15/5
I like that, since
cos^2 θ/2 = (1 + cosθ)/2 = (1 + 1/5)/2 = 3/5
But Ø/2 = pi - θ/2 so cos is negative.
For (b),
sin^2 θ/2 = (1 - cosθ)/2 = (1 - 1/5)/2 = 2/5
in QII, sin is positive, so sin Ø/2 = √(2/5) = √10/5
I assume 24/5 as you show is a typo, since that's greater than 1.
For (c), tan Ø/2 = sinØ/2 / cos Ø/2
= √(2/5)/-√(3/5)
= -√2/√3 = -√(2/3) or -√6/3
Apparently the book does not like radicals in the denominator, but they don't bother me.
cos a = 1/5
then sin a = sqrt (1-cos^2a) = sqrt(24/25)
cos(a/2) = -sqrt [(1+1/5)/2 ] = -sqrt (3/5)
So I agree with you.
To find the exact values of trigonometric functions, such as cos(1/2), sin(1/2), and tan(1/2), we can use the half-angle identities for trigonometric functions. These identities express the trigonometric functions of an angle half their original size in terms of the original trigonometric functions.
Let's start with finding the exact value of cos(1/2), given that cos(x) = 1/5 for an angle x that is greater than 270 degrees but less than 360 degrees.
To use the half-angle identity for cos(x), we have:
cos(x/2) = ± √((1 + cos(x)) / 2)
Substituting x = 2 in the above formula, we get:
cos(1/2) = ± √((1 + cos(2)) / 2)
Since cos(2) = 1 - 2sin^2(1), we substitute it into the equation:
cos(1/2) = ± √((1 + (1 - 2sin^2(1))) / 2)
Next, we need to determine the value of sin(1) using the given information. Since cos^2(1) + sin^2(1) = 1, we can rearrange the equation to solve for sin(1):
sin^2(1) = 1 - cos^2(1)
Given that cos(1) = 1/5, we have:
sin^2(1) = 1 - (1/5)^2
= 1 - 1/25
= 24/25
Taking the square root of both sides, we find:
sin(1) = ± √(24/25) = ± √24/5
Plugging this into the equation for cos(1/2), we have:
cos(1/2) = ± √((1 + (1 - 2(sin^2(1)))) / 2)
= ± √((1 + (1 - 2((± √24/5)^2))) / 2)
= ± √((1 + (1 - (2 * 24/5))) / 2)
= ± √((1 + (1 - 48/5)) / 2)
= ± √((1 + (5 - 48)/5) / 2)
= ± √((1 + (-43/5)) / 2)
= ± √((-43/5 + 5/5) / 2)
= ± √((-38/5) / 2)
= ± √(-38/10)
= ± √(-19/5)
Hence, the exact value for cos(1/2) is ± √(-19/5) or ± √(19/5) with the negative square root values due to the angle being in the fourth quadrant (greater than 270 degrees but less than 360 degrees).
Similarly, you can use the half-angle identity for sin(x) to find the exact value of sin(1/2), and the half-angle identity for tan(x) to find the exact value of tan(1/2). The answers you mentioned, which are given in the book, are likely derived using these identities. However, it seems that there may be a mistake in your book since the results you obtained are correct (i.e., a) -√3/5, b) √24/5, and c) no solution).