The identities shown in my last post showed how measuring 𝜃 from the negative x-axis affected its trig functions. Today’s post shows how measuring 𝜃 from the y-axis affects its trig functions.
Consider the diagram below:
The angle 𝜃 is shown measured from both the x and the y axes. The angle 𝜃 measured from the y-axis is 𝜋/2 – 𝜃 when conventionally measured from the positive x-axis. Notice that the right triangles formed from both measurements are identical – just in different orientations. The x coordinate of the 𝜋/2 – 𝜃 angle on the unit circle, cos(𝜋/2 – 𝜃), is the same as the y coordinate of the 𝜃 angle, sin𝜃. Similarly the y coordinate of the 𝜋/2 – 𝜃 angle on the unit circle, sin(𝜋/2 – 𝜃), is the same as the x coordinate of the 𝜃 angle, cos𝜃. This illustrates the following identities:
cos(𝜋/2 – 𝜃) = sin𝜃
sin(𝜋/2 – 𝜃) = cos𝜃
tan(𝜋/2 – 𝜃) = sin(𝜋/2 – 𝜃)/cos(𝜋/2 – 𝜃) = cos𝜃/sin𝜃 = 1/tan𝜃 = cot𝜃
I’ve introduced another trig function here, the cotangent, abbreviated cot. The cotangent is the reciprocal of the tangent.
When solving equations involving trig functions (which we will eventually do), these and the following identities can be used to convert sines to cosines and vice versa.
Now let’s measure 𝜃 from the other side of the y-axis. This gives us the conventional angle 𝜋/2 + 𝜃:
The only change here is that the x coordinate on the unit circle is now negative. So the resulting identities are:
cos(𝜋/2 + 𝜃) = -sin𝜃
sin(𝜋/2 + 𝜃) = cos𝜃
tan(𝜋/2 + 𝜃) = sin(𝜋/2 + 𝜃)/cos(𝜋/2 + 𝜃) = cos𝜃/-sin𝜃 = -1/tan𝜃 = -cot𝜃
Now the angle to the negative y-axis is 3𝜋/2 (270°). An angle measured to the left from the negative y-axis is the conventional angle 3𝜋/2 – 𝜃. The coordinates of this angle on the unit circle are swapped and negative of the angle 𝜃 in the first quadrant. So the picture looks like this:
And the corresponding identities are:
cos(3𝜋/2 – 𝜃) = -sin𝜃
sin(3𝜋/2 – 𝜃) = -cos𝜃
tan(3𝜋/2 – 𝜃) = sin(3𝜋/2 – 𝜃)/cos(3𝜋/2 – 𝜃) = -cos𝜃/-sin𝜃 = 1/tan𝜃 = cot𝜃
I will leave it as an exercise for you to show that for an angle measured to the right of the negative y-axis, the corresponding identities are:
cos(3𝜋/2 + 𝜃) = sin𝜃
sin(3𝜋/2 + 𝜃) = -cos𝜃
tan(3𝜋/2 + 𝜃) = sin(3𝜋/2 + 𝜃)/cos(3𝜋/2 + 𝜃) = -cos𝜃/sin𝜃 = -1/tan𝜃 = -cot𝜃
Next time, I will use the identities shown so far in some example problems.