(a) Use the definitions of sine and cosine to derive the...

(a) Use the definitions of sine and cosine to derive the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$ (b) Use the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$ to derive the other Pythagorean identities, $1+\tan ^{2} \theta=\sec ^{2} \theta$ and $1+\cot ^{2} \theta=\csc ^{2} \theta$ Discuss how to remember these identities and other fundamental identitics.

(a) Use the definitions of sine and cosine to derive the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$
(b) Use the Pythagorean identity $\sin ^{2} \theta+\cos ^{2} \theta=1$
to derive the other Pythagorean identities, $1+\tan ^{2} \theta=\sec ^{2} \theta$ and $1+\cot ^{2} \theta=\csc ^{2} \theta$
Discuss how to remember these identities and other fundamental identitics.

Key Concepts

Mnemonic and Memorization Strategies

Mnemonic strategies assist in remembering fundamental trigonometric identities, reducing reliance on rote memorization. Techniques include visualizing the unit circle, using mnemonics like 'SOH CAH TOA' for right triangle relationships, and practicing the derivations so that the relationships become intuitive, rather than memorized in isolation.

Derivation of Related Pythagorean Identities

This involves algebraic manipulation of the primary identity. By dividing sin²? + cos²? = 1 by cos²?, one derives 1 + tan²? = sec²?, and similarly by dividing by sin²?, one obtains 1 + cot²? = csc²?. Understanding this process reinforces how interrelated the trigonometric functions are through their definitions and fundamental identities.

Pythagorean Identity

At the core of trigonometry, the Pythagorean identity arises from the equation of the unit circle (x² + y² = 1). By substituting sine for the y-coordinate and cosine for the x-coordinate, we arrive at sin²? + cos²? = 1. This identity is a foundational relationship that underpins many other trigonometric results.

Sine and Cosine Definitions

This concept refers to the interpretation of sine and cosine as the ratios derived from a right triangle or as coordinates on the unit circle. Specifically, sine is defined as the ratio of the opposite side to the hypotenuse, and cosine as the ratio of the adjacent side to the hypotenuse, corresponding to the y- and x-coordinates of a point on the unit circle, respectively.

Transcript

00:01 Okay, how can we use the definitions of sine and cosine to derive the pythagorean identity sine squared plus cosine squared equals 1? so for this, let's look at the unit circle.

00:12 Let's take this triangle right here.

00:16 Remember that on the unit circle, it's a circle of radius 1, so that's a 1 right there.

00:25 Now let's think about this as theta there.

00:29 It's a little hard to see there.

00:31 And this is our y value this is our x value according to the pythagorean theorem remember it's a right triangle it looks like this if that's a little easier to see x squared plus y squared should be equal to one can we find that in terms of sine and cosine cosine theta you're going to be equal to the adjacent which is x over the hypotenuse of one sine theta is going to be equal to the opposite, which is y, the hypotenuse, which is one.

01:10 Cosine theta equals x, sine theta, equals y.

01:18 So just plug these in here.

01:22 That's how we find that cosine squared theta plus sine squared theta equals.

01:30 That's how we prove that.

01:32 Let's use this to derive the other identities, tangent, secant, cotangent, co -tangent, co -sequent.

01:39 Okay.

01:41 So, let's just start with this here.

01:48 Theta equals 1.

01:50 To start, let's just divide everything by cosine squared theta.

01:56 We can do that as long as we divide each and every term by cosine squared.

02:02 So when we do that, you get 1 plus sine over cosine, is tangent squared theta.

02:12 1 over cosine is secant squared theta.

02:18 So there we go...

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