A coil of wire rotating in a magnetic field induces a coltage E=20sin((PIa/4) - (PI/2)). Use an identity to express this in terms of cos(PIa/4).
Types of Identities:
Double Angle, Half Angle, Sum and Difference of Sine, Cosine, and Tangent, Pythagorean, Reciprocal, Quotient.
Thanks
PIa = pi*a
PI/2 = pi/2
E=-20cos(pi*a/4)
Any chance you could show a bit of work? Thanks.
sin(x-y)=sin(x)cos(y)-sin(y)cos(x)
Let y=pi/2
sin(x-pi/2)=sin(x)*0-1*cos(x)=-cos(x)
To express the given equation in terms of cos(PIa/4), we can use the sum and difference of sine identities along with the Pythagorean identity.
The sum and difference of sine identities state:
- sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
The Pythagorean identity states:
- sin^2(θ) + cos^2(θ) = 1
Let's use these identities to express E=20sin((PIa/4) - (PI/2)) in terms of cos(PIa/4):
1. Start with the given equation:
E = 20sin((PIa/4) - (PI/2))
2. Use the difference of sine identity:
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
Now substitute (A = PIa/4) and (B = PI/2):
E = 20sin(PIa/4)cos(PI/2) - 20cos(PIa/4)sin(PI/2)
3. Apply the Pythagorean identity to the cos(PI/2) term:
cos(PI/2) = √(1 - sin^2(PI/2))
Simplify:
E = 20sin(PIa/4)√(1 - sin^2(PI/2)) - 20cos(PIa/4)sin(PI/2)
4. Use sin(PI/2) = 1:
E = 20sin(PIa/4)√(1 - 1) - 20cos(PIa/4)(1)
Simplify further:
E = 20sin(PIa/4)(0) - 20cos(PIa/4)
5. Finally, simplify:
E = -20cos(PIa/4)
Therefore, the voltage E=20sin((PIa/4) - (PI/2)) can be expressed in terms of cos(PIa/4) as E = -20cos(PIa/4).