How Are the Trigonometric Identities for Cosine and Sine Related?

[nesta]

Hi friends,

Please make me understand this simplest function,

y = cos θ

2. in the next step it says: cos θ = sin (π/2 - θ)
3. and similarly -cos (π/2 - θ) = -sin θ.

Can anyone please explain how the steps 2 & 3 are deduced.

Thanks,
Nesta

 

[rochfor1]

Do you know what the unit circle is?

 

[HallsofIvy]

Hi friends,

Please make me understand this simplest function,

y = cos θ

2. in the next step it says: cos θ = sin (π/2 - θ)
3. and similarly -cos (π/2 - θ) = -sin θ.

Can anyone please explain how the steps 2 & 3 are deduced.

Thanks,
Nesta

The most basic definition of "cosine" is that it is "near side divided by hypotenuse" in a right triangle and of "sine" that it is "opposite side divided by hypotenuse".
Since a right triangle has one angle of size 90 degrees or $\pi/2$ radians, and the angles in any triangle sum to $\pi$ radians, the two acute angles must sum to $\pi- \pi/2= \pi/2$. That is, if one of the acute angles is $\theta$, then the other is $\pi/2- \theta$. And, of course, switching angles swaps "near" and "opposite" sides.

For a more general definition, where $\theta$ is not restricted to be between 0 and $\pi/2$ radians, you would have to go with something like the unit circle definition rochfor1 suggests.