# Trigonometric simplification: -a*sin(wt) + w*cos(wt)

• SubtleAphex
In summary, the equation b/(a^2+ ω^2 )*(-a*sin(ω*t)+ω*cos(ω*t)) can be simplified to (b/√(a^2+ ω^2 ))*sin(ω*t+ θ) where θ= tan^(-1) (- ω/a). This can be done using the sin sum formula or by writing it in terms of complex exponentials. The expression for θ may not be the principal value due to its location in the second quadrant.
SubtleAphex
I'm trying to figure out the steps required to do the following simplification:

This equation:
b/(a^2+ ω^2 )*(-a*sin(ω*t)+ω*cos(ω*t))

can be simplified to the following:
(b/√(a^2+ ω^2 ))*sin(ω*t+ θ)
θ= tan^(-1) (- ω/a)

I can numerically verify that this is true but I am having trouble figuring out the steps to do this simplification. Any help would be greatly appreciated. See attached .doc file for a cleaner presentation of the equations.

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• Simplification.doc
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SubtleAphex said:
I'm trying to figure out the steps required to do the following simplification:

This equation:
b/(a^2+ ω^2 )*(-a*sin(ω*t)+ω*cos(ω*t))

can be simplified to the following:
(b/√(a^2+ ω^2 ))*sin(ω*t+ θ)
θ= tan^(-1) (- ω/a)

I can numerically verify that this is true but I am having trouble figuring out the steps to do this simplification. Any help would be greatly appreciated. See attached .doc file for a cleaner presentation of the equations.

For this problem use the sin sum formula sin(ω*t+θ)=cos(θ)sin(ω*t)+sin(θ)cos(ω*t), where xcos(θ)=-a and xsin(θ)=ω. You then need to to solve for x and θ.

The other way to solve it, and many similar problems, is to write it in terms of complex exponentials. This isn't really any simpler, but it eliminates the need to memorize a million trig identities.

SubtleAphex said:
I'm trying to figure out the steps required to do the following simplification:

This equation:
b/(a^2+ ω^2 )*(-a*sin(ω*t)+ω*cos(ω*t))

can be simplified to the following:
(b/√(a^2+ ω^2 ))*sin(ω*t+ θ)
θ= tan^(-1) (- ω/a)

I can numerically verify that this is true but I am having trouble figuring out the steps to do this simplification. Any help would be greatly appreciated. See attached .doc file for a cleaner presentation of the equations.

Write it like this:

$$\frac{b}{a^2+\omega^2}\left(-a\sin(\omega t)+\omega\sin(\omega t)\right) = \frac b {\sqrt{a^2+\omega^2}}\left(\frac{-a}{\sqrt{a^2+\omega^2}}\sin(\omega t) +\frac{\omega}{\sqrt{a^2+\omega^2}}\cos(\omega t)\right)$$

Now, if you draw an angle in the second quadrant whose tangent is -ω/a you will see that the above expression becomes

$$\frac b {\sqrt{a^2+\omega^2}}\left(cos(\theta)\sin(\omega t) +\sin(\theta)\cos(\omega t)\right)=\frac b {\sqrt{a^2+\omega^2}}\sin(\omega t+\theta)$$

My only quarrel with the formula is that θ, being in the second quadrant, is not the principal value, which may not matter.

To simplify this trigonometric expression, we can use the following steps:

1. Use the trigonometric identity sin(A+B) = sin(A)cos(B) + cos(A)sin(B) to rewrite the expression as:

b/(a^2+ ω^2 )*(-a*sin(ω*t)+ω*cos(ω*t)) = (b/a^2+ ω^2 )*(-a*sin(ω*t)*cos(ω*t)+ω*cos^2(ω*t) + ω*sin(ω*t)*cos(ω*t))

2. Simplify the expression inside the parentheses using the identity cos^2(x) + sin^2(x) = 1:

(a^2+ ω^2 )*(-a*sin(ω*t)*cos(ω*t)+ω*cos^2(ω*t) + ω*sin(ω*t)*cos(ω*t)) = (a^2+ ω^2 )*(-a*sin(ω*t)*cos(ω*t)+ω*(1-sin^2(ω*t)) + ω*sin(ω*t)*cos(ω*t))

3. Distribute the (a^2+ ω^2) term and simplify:

(a^2+ ω^2 )*(-a*sin(ω*t)*cos(ω*t)+ω*(1-sin^2(ω*t)) + ω*sin(ω*t)*cos(ω*t)) = -a^2*sin(ω*t)*cos(ω*t)+ω^2+ ω^2*sin(ω*t)*cos(ω*t)

4. Use the identity sin(2x) = 2sin(x)cos(x) to simplify the expression further:

-a^2*sin(ω*t)*cos(ω*t)+ω^2+ ω^2*sin(ω*t)*cos(ω*t) = -a^2*sin(2ω*t)/2 + ω^2 + ω^2*sin(2ω*t)/2

5. Simplify the fractions and combine like terms:

-a^2*sin(2ω*t)/2 + ω^2 + ω^2*sin(2ω*t)/2 = (-a^2/2 + ω^2/2 + ω^2/2)*sin(2ω*t) = (b/√(a^2+ ω^2 ))*sin

## 1. What is "Trigonometric simplification"?

Trigonometric simplification is the process of reducing expressions involving trigonometric functions to their simplest form. This is done by using trigonometric identities and properties.

## 2. What does the expression -a*sin(wt) + w*cos(wt) mean?

The expression -a*sin(wt) + w*cos(wt) represents a combination of a sine and cosine function, where a and w are constants and t is the variable. This type of expression is often used in physics and engineering to model oscillatory motion.

## 3. How do you simplify -a*sin(wt) + w*cos(wt)?

To simplify -a*sin(wt) + w*cos(wt), we can use the trigonometric identity cos(x)sin(y) + sin(x)cos(y) = sin(x+y). This gives us a single sine term with a coefficient of a+w, making the expression -a*sin(wt) + w*cos(wt) = (a+w)*sin(wt).

## 4. Can -a*sin(wt) + w*cos(wt) be simplified further?

No, -a*sin(wt) + w*cos(wt) is already in its simplest form. It cannot be simplified any further using trigonometric identities or properties.

## 5. In what situations would I need to use trigonometric simplification?

Trigonometric simplification is commonly used in mathematical modeling, physics, and engineering to simplify expressions involving trigonometric functions. It is also useful in solving trigonometric equations and evaluating integrals involving trigonometric functions.

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