Problems 65-68 are based on material learned earlier in the course. The purpose of these problem

Problems 65-68 are based on material learned earlier in the...

Key Concepts

Half-Angle Formulas

Half-angle formulas provide a way to express trigonometric functions of half an angle in terms of the trigonometric functions of the original angle. They allow one to determine values like sin(?/2), cos(?/2), and tan(?/2) using expressions that involve cos(?) and sin(?), which is particularly useful when the original angle's trigonometric values are known.

Pythagorean Identity

The Pythagorean identity, expressed as sin²? + cos²? = 1, is a fundamental relationship in trigonometry that connects the sine and cosine of an angle. This identity is frequently used to deduce the sine of an angle when the cosine is given or vice versa. In the context of half-angle problems, it helps in obtaining missing trigonometric values needed in half-angle formulas.

Quadrant Analysis

Quadrant analysis involves determining the appropriate sign (positive or negative) of trigonometric function values based on the angle's position in the unit circle. This is crucial when applying half-angle formulas, as the formulas yield square roots, and the correct sign must be chosen to reflect the true value of the function in the given interval.

Transcript

00:01 Okay, so we're given that the cosine of this angle alpha is four -fifths, and our angle alpha is between zero and pi over two.

00:11 That means it's in the first quadrant.

00:17 We want to remember that because it'll be important later.

00:20 So the first thing we want to find, we want to find cosine of alpha, that kind of looks like an x.

00:30 Cosine of alpha over two.

00:35 Okay? so this is where we're going to have to use our half -angle identities.

00:41 And if you remember, the half -angle identity for cosine is plus or minus the square root of one plus cosine alpha over two.

01:00 Okay, so we need to figure out first if it's positive or negative.

01:07 Well, since we're in quadrant one, we know cosine is positive in quadrant one.

01:13 So we're going to choose positive.

01:16 Okay, now we can just plug in what we have for cosine of alpha.

01:21 So that gives us a square root of 1 plus cosine of alpha, which is 4 over 5, all over two.

01:36 Okay? now i need to simplify.

01:39 So let's simplify the top first.

01:44 One is 5 over 5.

01:47 So we have 5 over 5 plus 4 over 5 over 2.

01:55 Okay? so that lets us see that the top is 9 over 5.

02:04 9 over 2.

02:07 Okay.

02:08 Well, this is still pretty complicated.

02:09 We have a fraction on top in the new.

02:11 Of another fraction.

02:13 So to simplify that we can use our rule for fractions saying that the top is multiplied by the reciprocal of the bottom.

02:25 So i'll show you what i mean.

02:27 This equals a square root of nine -fifths times one -half.

02:38 Okay, and that simplifies to the square root of nine over ten.

02:46 Okay, so that's what cosine of alpha over two is.

02:55 Box that in a little better.

03:01 Okay, now the next thing we want to find is sine of alpha over two.

03:13 Okay, well, we'll do the same thing.

03:15 We'll use our half -angle identity for sign this time, and our half -angle identity for sign is plus or minus the square root of one minus co -scent.

03:30 That's the difference between sine and cosine is a minus here instead of a plus over two okay so first things first again we'll figure out whether we want a positive or a negative because we're in quadrant one sign is positive in quadrant one so we are positive okay so now let's plug in cosine of alpha we have one minus four over five over two all inside the square root.

04:07 Okay? one is 5 over 5.

04:10 So 5 over 5 minus 4 over 5 is 1 over 5.

04:21 Okay? and if we do our same trick up here, we see that we have the square root of 1 over 5 times one half.

04:34 And that is the square root of one tenth...

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