We are asked to find trigonometric identities that are not derived from the Pythagorean Identity. Let's take a look at the graph of the sine and cosine functions. Notice that we can match both graphs by translating the cosine function π2 units to the right.
This means that the translation by π2 units to the right of the cosine function is identical to the sine function.
cos(θ-π/2)=sinθThe graphs can also be matched by translating the sine function π2 units to the left.
This means that the translation by π2 units to the left of the sine function is identical to the cosine function.
sin(θ+π/2)=cosθ
Let's now take into account the symmetry of the sine and cosine functions. We begin by reviewing the two types of symmetry in functions.
Symmetry of Functions
Odd Symmetry
f(- x)= - f(x)
Even Symmetry
f(- x) = f(x)
Let's take another look at the sine and cosine functions.
We notice that the sine function has odd symmetry and the cosine function has even symmetry. This means that we can write two more identities.
sin(- θ) = - sinθ cos(- θ) = cosθ
Finally, we will summarize our findings.
Trigonometric Identities
cos(θ-π/2)=sinθ
sin(θ+π/2)=cosθ
sin(- θ) = - sinθ
cos(- θ) = cosθ
Please note that these are just some of the many other trigonometric identities. We can even combine the above identities to get new ones!
