In this discussion, we are going use the sum-to-product identities to solve trig equations
I have already proved the sum-to-product identities in a previous discussion, so check it out if you are interested!
Let’s begin!
Let’s say you have the equation
sin(2x) + sin(x) = 0 , [0,2pi)
We want to solve for x on the interval [0,2pi)
We know by the sum-to-product identity that:
sun(a) + sin(b) = 2sin[(a+b)/2)cos[(a-b)/2]
Thus,
sin(2x) + sin(x) = 0
2sin[(2x+x)/2)]cos[(2x-x)/2]=0
2sin[3x/2]cos[x/2]=0
sin[3x/2]cos[x/2]=0
Now let A = 3x/2 and B= x/2.
Then we have,
sin[A]cos[B]=0, which means that,
sin[A]=0 and cos[B]=0.
Thus,
A = 0 + 2pi*n
A = pi + 2pi*n
B = pi/2 + 2pi*n
B = 3pi/2 + 2pi*n, where n is an integer.
Replacing A and B, we have,
3x/2 = 0 + 2pi*n
3x/2 = pi + 2pi*n
x/2 = pi/2 + 2pi*n
x/2 = 3pi/2 + 2pi*n, where n is an integer.
For the first equation, we solve for x and plug in all values for n.
3x/2 = 0 + 2pi*n
3x= 4pi*n
x= (4pi*n)/3
Plugging in values for n, we have:
x = 0, when n=0
x= 4pi/3, when n= 1
All integers greater than 1 will go outside of the interval [0,2pi).
For the second equation, we solve for x and plug in all values for n.
3x/2 = pi + 2pi*n
3x= 2pi + 4pi*n
x= 2pi/3 + (4pi*n)/3
Plugging in values for n, we have:
x = 2pi/3, when n=0
All integers greater than zero will go outside of the interval [0,2pi).
For the third equation, we solve for x and plug in all values for n.
x/2 = pi/2+ 2pi*n
x= pi + 4pi*n
Plugging in values for n, we have:
x= pi, when n= 0
All integers greater than 0 will go outside of the interval [0,2pi).
For the fourth equation, we solve for x and plug in all values for n.
x/2 = 3pi/2+ 2pi*n
x= 3pi + 4pi*n
There are no values for x because all integers go outside of the interval [0,2pi).
Thus, the solutions for x are
x= 0, 4pi/3, 2pi/3, and pi.
Thank you for reading/watching!
I hope that I can assist you on your math journey!
Sincerely,
➕MathAngel369➕
🎥 Suggested Video and Playlist:
Video: Proof of the Sum-to-Product Formulas for Sine
Video: Proof of the Product-to-Sum Formulas for Cosine
Playlist: Proving/Verifying and Using Trig Identities
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