Trigonometry, Part 2

Now let’s use the unit circle to see some of the common trig identities. These identities (rules) will be used in future posts.

Let’s assume we have an acute angle 𝜃. An acute angle is one that is between 0 and 𝜋/2 (or 0 to 90°). The following identities are valid for any angle, not just acute ones – it is just easier to see the logic in the diagram if we assume this.

The following picture shows the relationship between an angle 𝜃 in the first quadrant, and an angle in the second quadrant which is symmetric with 𝜃:

You can see that to measure this symmetric angle from the postive x-axis, you just subtract it from 𝜋. The coordinates of the intersected point on the unit circle are negative for the x coordinate but the same y coordinate as the original angle 𝜃. So the following identities are evident from this picture:

cos(𝜋 – 𝜃) = -cos𝜃
sin(𝜋 – 𝜃) = sin𝜃
tan(𝜋 – 𝜃) = -cos𝜃/sin𝜃 = -tan𝜃

Again, these are true for any angle, not just acute ones.

As an example, let 𝜃 = 𝜋/3, (60°). The following is true for 𝜋/3:

cos(𝜋/3) = 1/2
sin(𝜋/3) = √3̅/2
tan(𝜋/3) = √3̅

Now 𝜋 – 𝜋/3 = 2𝜋/3. So using these identities, we know that

cos(2𝜋/3) = -1/2
sin(2𝜋/3) = √3̅/2
tan(2𝜋/3) = -√3̅

Now let’s look at a symmetric angle in the third quadrant. To measure this angle from the positive x-axis, you add it to 𝜋. The corresponding coordinates of the intersected point on the unit circle are both the negative of the coordinates for 𝜃. So the following identities are shown in this picture:

So these identities are

cos(𝜋 + 𝜃) = -cos𝜃
sin(𝜋 + 𝜃) = -sin𝜃
tan(𝜋 + 𝜃) = -cos𝜃/-sin𝜃 = tan𝜃

Using our same example, 𝜋 + 𝜋/3 = 4𝜋/3. Using these identities:

cos(4𝜋/3) = -1/2
sin(4𝜋/3) = -√3̅/2
tan(4𝜋/3) = √3̅

One more quadrant to go:

As was mentioned before, angles measured clockwise from the positive x-axis are negative. So the following trig identities are shown in the figure above:

cos(-𝜃) = cos𝜃
sin(-𝜃) = -sin𝜃
tan(-𝜃) = cos𝜃/-sin𝜃 = -tan𝜃

So,

cos(-𝜋/3) = 1/2
sin(-𝜋/3) = -√3̅/2
tan(-𝜋/3) = -√3̅

There are a couple more identities I would like to show but I’ll save that for next time.