In Exercises 31-34 , match each graph with a pair of the...

In Exercises $31-34$ , match each graph with a pair of the following equations. Use your knowledge of identities and transformations, not your grapher. (a) $y=\cos (3-2 x)$ (b) $y=\sin x \cos 1+\cos x \sin 1$ (c) $y=\cos (x-3)$ (d) $y=\sin (2 x-5)$ (e) $y=\cos x \cos 3+\sin x \sin 3$ (f) $y=\sin (x+1)$ (f) $y=\sin (x+1)$ (g) $y=\sin (x+1)$ (g) $y=\cos 3 \cos 2 x+\sin 3 \sin 2 x$ (h) $y=\sin 2 x \cos 5-\cos 2 x \sin 5$ $$[-2 \pi, 2 \pi]$ by $[-1,1]$$

In Exercises $31-34$ , match each graph with a pair of the following equations. Use your knowledge of identities and transformations, not your grapher.
(a) $y=\cos (3-2 x)$
(b) $y=\sin x \cos 1+\cos x \sin 1$
(c) $y=\cos (x-3)$
(d) $y=\sin (2 x-5)$
(e) $y=\cos x \cos 3+\sin x \sin 3$
(f) $y=\sin (x+1)$
(f) $y=\sin (x+1)$
(g) $y=\sin (x+1)$
(g) $y=\cos 3 \cos 2 x+\sin 3 \sin 2 x$
(h) $y=\sin 2 x \cos 5-\cos 2 x \sin 5$

$$[-2 \pi, 2 \pi]$ by $[-1,1]$$

Key Concepts

Trigonometric Addition Formulas

These formulas, such as cos(A - B) = cos A cos B + sin A sin B and sin(A + B) = sin A cos B + cos A sin B, are fundamental tools for rewriting expressions involving sums or differences of angles into a single trigonometric function. Mastery of these identities is crucial for recognizing when different-looking expressions are actually equivalent, which is often the key step in matching algebraic expressions to their graphs.

Graph Transformations of Trigonometric Functions

Graph transformations include adjustments such as shifting (phase shift), stretching or compressing (vertical/horizontal scaling), and reflecting a function. Understanding how these alterations affect the standard sine and cosine graphs allows one to interpret and match equations, even when they have been transformed by changes inside the function's argument or factors multiplying the function.

Even and Odd Properties of Trigonometric Functions

Recognizing that cosine is an even function (cos(-?) = cos ?) and sine is an odd function (sin(-?) = - sin ?) helps simplify and compare different trigonometric expressions. These properties are particularly useful when rewriting trigonometric equations into standard forms, thereby facilitating the process of matching them to their corresponding graphs.

Trigonometric Identities

A thorough knowledge of various trigonometric identities, not limited to the angle addition formulas, is essential for working with and transforming trigonometric expressions. These identities allow the conversion of complex expressions into simpler forms, making it easier to understand the underlying structure of the function and accurately predict its graphical behavior.

Transcript

00:01 Number 31 asks you to match the given graph with a pair of suggested equations.

00:16 So i would like you first to use, and it also suggests that you use your knowledge of identities and transformations.

00:24 Don't use a grapher.

00:27 So here, i'm going to first appeal to your knowledge of transformations.

00:35 When you look at number 31, we evidently are looking for cosine or sign curves, right? sign or cosine equations.

00:51 When i first look at 31, i see a sign curve.

00:58 I want to encourage you to see the sign curve as well.

01:02 I see a sign curve that starts less than 90 degrees before zero.

01:21 This is what i see.

01:23 This would be 90 degrees or pi over 2.

01:26 And the curve is starting here, as imperfect as this is.

01:32 The curve is starting there.

01:36 It's less than 90 degrees.

01:39 Because of that, we know that the phase shift here is going to have a positive, right? because it shifted left.

01:50 And when i look at the options here, i see that d has sign of 2x minus 5.

01:57 That would give us, so if we worked with, i want to show you this, the sign of 2x minus 5, that would give us a phase shift of negative 5 over 2, which that would be a positive shift to the right anyway...

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