Where Do We Use This: Homework Statement

[Endorser]

Homework Statement

I have an assignment that is based on the forumal

A sin x + B cos x = K sin(x+ φ).

I need to: 1.Explain why we might need such a formula
2. Show proof or dervivation of this formula on which you add in english the
reasons for each step. (I know how to go from A sin x + B cos x to K sin(x+ φ))
3.Describe a real application of this formula

Homework Equations

A sin x + B cos x = K (sin x cos φ + cos x sin φ) = K sin(x+ φ).

The Attempt at a Solution

I have done a lot of searching and have come up with nothing...
Any help is appreciated

Thanks

 

[Mindscrape]

You probably ought to explicitly say the relation between A and B to K and Phi.

This kind of formula is mostly used for simple harmonic motion: if you have a mass on a spring or a pendulum swinging at small angles. It's actually more typical to see the formula as a cos argument rather than a sine.

 

[Endorser]

You probably ought to explicitly say the relation between A and B to K and Phi.

Could you explain to me what relation they hold ?

 

[Mindscrape]

A = K cos φ
B = K sin φ

comes directly from your second equation, the one with the trig identity.

 

[rwisz]

1. The formula A sin x + B cos x = K sin(x+ φ) is known as the trigonometric identity for the sum of two angles. We might need this formula in various situations, such as solving equations involving trigonometric functions, evaluating integrals, and representing periodic functions in terms of sine and cosine functions.

2. The proof of this formula can be done using basic trigonometric identities and the concept of angle sum and difference formulas. Here is a step-by-step explanation:

Step 1: Start with the left-hand side of the equation A sin x + B cos x. Using the angle sum formula for sine, we can rewrite this as A(sin x cos φ + cos x sin φ).

Step 2: Use the same angle sum formula for cosine to rewrite B cos x as B(cos x cos φ - sin x sin φ).

Step 3: Combine the two terms by factoring out the common factor of cos φ, which gives us A sin x cos φ + B cos x cos φ.

Step 4: Use the trigonometric identity sin²x + cos²x = 1 to replace sin²x with (1-cos²x). This gives us A(1-cos²x)cos φ + B cos x cos φ.

Step 5: Distribute the terms and rearrange them to get Acos φ - Acos²x cos φ + B cos x cos φ.

Step 6: Use the trigonometric identity sin²x + cos²x = 1 again to replace cos²x with (1-sin²x). This gives us Acos φ - A(1-sin²x)cos φ + B cos x cos φ.

Step 7: Distribute the terms and rearrange them to get A cos φ - A cos φ sin²x + B cos x cos φ.

Step 8: Factor out cos φ from the first two terms and sin φ from the last two terms, which gives us A cos φ (1-sin²x) + B sin φ cos x.

Step 9: Use the trigonometric identity sin²x + cos²x = 1 again to replace (1-sin²x) with cos²x. This gives us A cos φ cos²x + B sin φ cos x.

Step 10: Use the trigonometric identity cos²x = 1 - sin²x to rewrite