The above is wrong. (A - B)2 ##\neq## A2 - B2.NanoChrisK said:Homework Statement
Does [sin(θ/2)-cos(θ/2)]^2 equal 1 for all values of θ?
I need to figure this out to solve a physics problem.
Homework Equations
sin^2(A)+cos^2(A)=1
The Attempt at a Solution
[sin(θ/2)-cos(θ/2)]^2
=sin^2(θ/2)+cos^2(θ/2)
NanoChrisK said:=1
But this isn't what I get when I put f(x)=[sin(θ/2)-cos(θ/2)]^2 in my graphing calculator. What's going on?
Mark44 said:The above is wrong. (A - B)2 ≠\neq A2 - B2.
Mark44 said:The above is wrong. (A - B)2 ##\neq## A2 - B2.
No, this part is wrong: A2 + AB - AB + (-B)2, namely the +AB term.NanoChrisK said:Hmm I may be wrong, but I think you meant to type "(A -B)2 ≠ A2+A2" (with a + in the second half of the equation) This is the form I gave it in, unless I'm missing something. Is this not correct? I'm getting this from:
(A-B)2 = (A-B)(A-B) = A2 + AB - AB + (-B)2 = A2 + B2
Mark44 said:No, this part is wrong: A2 + AB - AB + (-B)2, namely the +AB term.
Follows from ##cos {2θ}=cos^2 θ - sin^2θ##...HallsofIvy said:It is fairly easy to show that [itex]cos^2(\theta/2)- sin^2(\theta/2)= cos(\theta)[/itex], not 1.
The equation represents the Pythagorean identity, which is a fundamental trigonometric identity that relates the three basic trigonometric functions: sine, cosine, and tangent. It states that for any angle θ, the square of the sine of half the angle plus the square of the cosine of half the angle equals 1.
The equation is important because it allows us to easily express and manipulate trigonometric expressions in terms of sine and cosine. It is also used in many applications involving triangles, circles, and periodic functions.
The equation is closely related to the unit circle, which is a circle with a radius of 1 centered at the origin on a coordinate plane. The Pythagorean identity can be derived from the unit circle, as the sine and cosine functions are defined as the y and x coordinates, respectively, of a point on the unit circle.
Yes, the Pythagorean identity can be used to solve trigonometric equations by manipulating the equation to isolate the unknown variable. It can also be used to simplify complex trigonometric expressions and verify trigonometric identities.
Yes, there are several other variations of the Pythagorean identity, such as Sin^2(θ)+cos^2(θ)=1, 1+tan^2(θ)=sec^2(θ), and 1+cot^2(θ)=csc^2(θ). These identities all relate the three basic trigonometric functions and are used in various contexts in mathematics and physics.