Express 3sin(3x)-4cos(3x) in the form Rcos(3x+\alpha)

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In summary, the problem is to express 3sin(3x)-4cos(3x) in the form Rcos(3x+\alpha),\alpha\ge0;R>0 and then find the smallest possible value of x for which 3sin(3x)-4cos(3x)=4. This can be solved using the identity a~cos(3x) + b~sin(3x) = \sqrt{a^2 + b^2}~cos \left ( 3x - tan^{-1} \left ( \frac{b}{a} \right ) \right ) where a = 3, b = -4 and x > 0. More information and
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ThomsonKevin
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Tried simplifying it of course, but didn't get far. Here's tbe problem:

''Express 3sin(3x)-4cos(3x) in the form Rcos(3x+\alpha),\alpha\ge0;R>0. Hence, find the smallest possible value of x for which 3sin(3x)-4cos(3x)=4.''

Bit confusing for me, especially the last part. How do you solve this, lads?
 
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  • #2
ThomsonKevin said:
Tried simplifying it of course, but didn't get far. Here's tbe problem:

''Express 3sin(3x)-4cos(3x) in the form Rcos(3x+\alpha),\alpha\ge0;R>0. Hence, find the smallest possible value of x for which 3sin(3x)-4cos(3x)=4.''

Bit confusing for me, especially the last part. How do you solve this, lads?
\(\displaystyle a~cos(3x) + b~sin(3x) = \sqrt{a^2 + b^2}~cos \left ( 3x - tan^{-1} \left ( \frac{b}{a} \right ) \right )\)
where a = 3, b = -4. (Warning: We have to take x > 0 for this.)

You can find that identity (and many others) here.

-Dan
 

1. What is the purpose of expressing an equation in the form Rcos(3x + α)?

The purpose of expressing an equation in the form Rcos(3x + α) is to simplify and condense the original equation. This form allows for easier identification of the amplitude, period, and phase shift of the trigonometric function, making it easier to analyze and understand the behavior of the function.

2. How do you determine the values of R and α in the equation Rcos(3x + α)?

To determine the values of R and α, you need to use the trigonometric identities of cosine and sine. In the given equation, R represents the amplitude of the function, which can be found by taking the square root of the sum of the squares of the coefficients of sin(3x) and cos(3x). α represents the phase shift, which can be found by taking the inverse tangent of the coefficient of sin(3x) divided by the coefficient of cos(3x).

3. Can you express any trigonometric function in the form Rcos(3x + α)?

Yes, any trigonometric function can be expressed in the form Rcos(3x + α). This form is derived from the cosine function, which is a fundamental trigonometric function and can be used to represent all other trigonometric functions.

4. What is the significance of the coefficient 3 in the equation Rcos(3x + α)?

The coefficient 3 in the equation Rcos(3x + α) represents the frequency of the trigonometric function. It determines the number of cycles the function completes in a given interval. In this case, the function completes 3 cycles in the interval of 2π, which means it has a frequency of 3 cycles per 2π units.

5. How can expressing an equation in the form Rcos(3x + α) be useful in real-life applications?

Expressing an equation in the form Rcos(3x + α) can be useful in real-life applications where periodic functions are involved, such as in sound and light waves. This form allows for easier analysis and prediction of the behavior of these waves, which can be applied in fields such as music, acoustics, and optics.

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