Determining Values On The Unit Circle

For the evaluation of cosΘ and sinΘ, we will use the following equivalencies:

(1)
\begin{align} x = cos(\theta) \end{align}
(2)
\begin{align} y = sin(\theta) \end{align}
(3)
\begin{align} \frac{y}{x} = \frac{sin(\theta)}{cos(\theta)} = tan(\theta) \end{align}

# Evaluating CosΘ and SinΘ at Θ = 0, 90, 180, 270, and 360 Deg.

Fortunately, evaluating Θ = 0, 90, 180, 270, and 360 degrees is simple because these points lie either directly on the x-axis or the y-axis. The following diagram illustrates this thusly.

Using the equivalencies at the beginning of this lesson, we notice that we can obtain the values of cosΘ and sinΘ for these special angles by looking at the (x , y) coordinates of the ray S intersecting the unit circle established by said angles. Fortunately, these coordinates are easy to acknowledge. The table below summarizes these special angles:

Θ in Degrees Θ in Radians CosΘ SinΘ
Θ = 0 deg. Θ = 0 rad. CosΘ = 1 SinΘ = 0
Θ = 90 deg. Θ = π/2 rad. CosΘ = 0 SinΘ = 1
Θ = 180 deg. Θ = π rad. CosΘ = -1 SinΘ = 0
Θ = 270 deg. Θ = 3π/2 rad. CosΘ = 0 SinΘ = -1
Θ = 360 deg. Θ = 2π rad. CosΘ = 1 SinΘ = 0

# Evaluating CosΘ and SinΘ at Θ = 30, 45, 60 Deg.

Evaluating Θ = 30, 45, 60 degrees is a little more difficult. Here is a simply trick to quickly construct a visual representation of the values of cosΘ and sinΘ in quadrant 1.

1. First construct a unit circle. Note that the diagram above only shows a portion of the unit circle for simplicity. We will now draw 3 different rays starting at the origin that approximate the angles of 30 degrees, 45 degrees, and 60 degrees. We will then set up pairs of coordinates.

As an intermediate acknowledgement, the term "abscissa" refers to the x-coordinate in an ordered pair, while the term "ordinate" refers to the y-coordinate in an ordered pair. Hence, for a point at (2, 3), the abscissa is 2 and the ordinate is 3.

2. Now we will start from the top. The first abscissa will be 1. The abscissa underneath it will be 2. And the abscissa at the bottom will be 3. We will then start back up again and have the topmost ordinate be 3. The middle ordinate will be 2. And the ordinate at the bottom will be 1.

3. We will now divide all of the the abscissas and ordinates by 2.

4. The final step is to squareroot the numerators of all of the abcissas and ordinates. We now appropriately have the values of cosΘ and sinΘ evaluated at Θ = 30, 45, and 60 degrees. The table below summarizes these angles.

Θ in Degrees Θ in Radians CosΘ SinΘ
Θ = 30 deg. Θ = π/6 rad. CosΘ = √3/2 SinΘ = 1/2
Θ = 45 deg. Θ = π/4 rad. CosΘ = √2/2 SinΘ = √2/2
Θ = 60 deg. Θ = π/3 rad. CosΘ = 1/2 SinΘ = √3/2

Be aware that you may see some various forms for some of the values on the unit circle such as the value at Θ = 45 degrees. Just acknowledge that:

(4)
\begin{align} \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}} \end{align}

# Evaluating TanΘ at Θ = 0, 90, 180, 270, and 360 Deg.

We have the following relationship for tanΘ such that:

(5)
\begin{align} \frac{y}{x} = \frac{sin(\theta)}{cos(\theta)} = tan(\theta) \end{align}

And from earlier in this lesson, we established a table to acknowledge from the unit circle for finding values of cosΘ and sinΘ given Θ = 0, 90, 180, 270, and 360. Thus we can evaluate tanΘ by using this equality. Note that over time you will not need to use the equality that relates tanΘ, cosΘ, and sinΘ as you begin to remember those values.

For example, let's say we want to find tan180. To do so, we know that cos180 = -1, and sin180 = 0. Thus we obtain:

(6)
\begin{align} tan(180^\circ) = \frac{sin(180^\circ)}{cos(180^\circ)} = \frac{0}{-1} = 0 \end{align}

Thus we obtain that tanΘ = 0.

## Undefined Values of TanΘ

Unfortunately, we cannot divide any number by 0. Hence, when cosΘ = 0, then tanΘ will be undefined. We notice from the table mentioned earlier that cosΘ = 0 when Θ = 90 degrees and 270 degrees. Hence:

(7)
\begin{align} tan(90^\circ) = \mathbf{undefined} \end{align}
(8)
\begin{align} tan(270^\circ) = \mathbf{undefined} \end{align}

The table below outlines the values of tanΘ when Θ = 0, 90, 180, 270, and 360 degrees.

Θ in Degrees Θ in Radians TanΘ
Θ = 0 deg. Θ = 0 rad. TanΘ = 0
Θ = 90 deg. Θ = π/2 rad. TanΘ = undefined
Θ = 180 deg. Θ = π rad. TanΘ = 0
Θ = 270 deg. Θ = 3π/2 rad. Tan Θ = undefined
Θ = 360 deg. Θ = 2π rad. TanΘ = 0

# Evaluating TanΘ at Θ = 30, 45, and 60 Deg.

We will bring back this relationship from the top of this lesson:

(9)
\begin{align} \frac{y}{x} = \frac{sin(\theta)}{cos(\theta)} = tan(\theta) \end{align}

If we are given two trigonometric identities, then we can always solve for the third trigonometric identity by isolation. Thus, it may be useful to recall what has been mentioned above in order to determine the value of tanΘ if forgotten.

For example, if we have Θ = 60 degrees and we want to determine the value of tanΘ, we can simply use the relationship that tanΘ is equivalent to sinΘ divided by cosΘ. Therefore:

(10)
\begin{align} tan(60^\circ) = \frac{sin(60^\circ)}{cos(60^\circ)} \end{align}

But we can evaluate the righthand side if we remember the values of sin60 and cos60. Thus we obtain:

(11)
\begin{align} tan(60^\circ) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = 2 \frac{\sqrt{3}}{2} = \sqrt{3} \end{align}

Thus:

(12)
\begin{align} tan(60^\circ) =\sqrt{3} \end{align}

The table below references the values of tanΘ when Θ = 30, 45, and 60 degrees.

Θ in Degrees Θ in Radians CosΘ
Θ = 30 deg. Θ = π/6 rad. TanΘ = 1/√3
Θ = 45 deg. Θ = π/4 rad. TanΘ = 1
Θ = 60 deg. Θ = π/3 rad. TanΘ = √3