Cast Rule

The CAST rule is a special rule used often as a memorization acronym in trigonometry. Essentially, "CAST" stands for COSINE-ALL-SINE,TANGENT. Starting in quadrant 4 and going in a counter clock-wise fashion, the CAST rule will tell us what trigonometric ratios (cosine, sine, and tangent) are always positive in said quadrants.

Pythagoras' Theorem

Before we continue on to define the CAST rule, it will be appropriate to review Pythagoras' Theorem. Pythagoras' Theorem is as follows for any triangle that has a right-angle (90 degree angle) within in. :

\begin{equation} a^2 + b^2 = c^2 \end{equation}

Where c is the hypotenuse (or the longest side on a right-angle triangle), and a and b are arbitrarily the other two sides.

Geometrically, the following diagram illustrates Pythagoras' Theorem:


Where the area of c-squared is equal to that of the area of a-squared plus b-squared for any right-angle triangle.

Nevertheless, this property will be important when we define the CAST rule.

Defining the CAST Rule

Remember that when defining the unit circle, we inscribed right-angle triangles defined by a portion of the x-axis and some ray, S and a perpendicular line from the termination of S to the x-axis? Our creation of that perpendicular line from the terminal of ray S to the x-axis means that we had essentially formed right-angle triangles. Hence, we can use Pythagoras' theorem effectively for defining the CAST rule. Let's first look at triangles we made in quadrant 1:


We notice that in quadrant 1 our x-values are restricted from 0 to ∞, while our y-values are also restricted from 0 to ∞. More precisely, our unit circle x-values are restricted from 0 to 1, while our y-values are also restricted from 0 to 1.

Thus for quadrant 1, we say that:

\begin{align} 0 < x_{Q1} < 1 \quad , \quad 0 < y_{Q1} < 1 \end{align}

But we know that:

\begin{align} x = cos(\theta) \quad , \quad y = sin(\theta) \end{align}

Thus we can obtain the inequality:

\begin{align} 0 < cos(\theta)_{Q1} < 1 \quad , \quad 0 < sin(\theta)_{Q1} < 1 \end{align}

So essentially, we know that in quadrant 1, cosΘ and sinΘ are positive in quadrants 1 since their values must both exist between 0 and 1 when we construct an inscribed triangle in quadrant 1.

However, we know that:

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

But since cosΘ and sinΘ are both positive in quadrant 1, then tanΘ must also be positive because tanΘ can be expressed in terms of cosΘ and sinΘ, thus we obtain:

\begin{align} 0 < tan(\theta)_{Q1} < 1 \end{align}

We will now look at triangles constructed in quadrant 2:


We notice that an angle inscribed within quadrants 2, the x-values are restricted between -∞ and 0, and the y-values are restricted between 0 and ∞. More precisely for our unit circle, the x-values are restricted between -1 and 0, and the y-values are restricted from 0 to ∞. Hence, we obtain:

\begin{align} -1 < x_{Q2} < 0 \quad , \quad 0 < y_{Q2} < 1 \end{align}
\begin{align} -1 < cos(\theta)_{Q2} < 0 \quad , \quad 0 < sin(\theta)_{Q2} < 1 \end{align}

Or more precisely cosΘ will be negative, and sinΘ will be positive in quadrant 2. However we know that tanΘ is represented in terms of cosΘ and sinΘ. In terms of just positive/negative signs we obtain:

\begin{align} tan(\theta) = \frac{sin(\theta)}{cos(\theta)} \end{align}
\begin{align} tan(\theta) = \frac{+}{-} \end{align}

Or more precisely, tanΘ will be negative in quadrant 2 always.

Now let's look at quadrant 3 from the following diagram below:


We that when a triangle constructed is in quadrant 3, then the x-values are restricted from -∞ to 0 and the y-values are restricted from -∞ to 0. More precisely for our unit circle, the x-values are restricted from -1 to 0, and the y-values are restricted from -1 to 0. Thus we obtain:

\begin{align} -1 < x_{Q3} < 0 \quad , \quad -1 < y_{Q3} < 0 \end{align}
\begin{align} -1 < cos(\theta)_{Q3} < 0 \quad , \quad -1 < sin(\theta)_{Q3} < 0 \end{align}
\begin{align} tan(\theta) = \frac{-}{-} \end{align}

Thus we obtain that cosΘ and sinΘ will always be negative when constructed in quadrant 3, while tanΘ will always be positive because a negative divided by a negative is a positive.

We will not go through quadrant 4, however, the procedure follows identically to that of quadrants 1, 2, and 3 we derived above. The table below will summarize the signs (positive or negative) for cosine, sine, and tangent:

CosΘ SinΘ TanΘ
Quadrant 1 Positive Positive Positive
Quadrant 2 Negative Positive Negative
Quadrant 3 Negative Negatives Positive
Quadrant 4 Positive Negatives Negative

From this diagram, we can determine that cosΘ is positive in quadrants 1 and 4, sinΘ is positive in quadrants 1 and 2, and tanΘ is positive in quadrants 1 and 3. More intuitively, cosΘ is exclusively positive in quadrant 4, sinΘ is exclusively positive in quadrant 2, and tanΘ is exclusively positive in quadrant 3. The table below summarizes as follows:

cell-content CosΘ SinΘ TanΘ
Exclusively Positive in: Quadrant 4 Quadrant 2 Quadrant 3

Thus this is how we define the CAST Rule:


Where the letters of CAST starting at quadrant 4 going counter clockwise state which trigonometric ratios are exclusively positive in that quadrant.

Evaluating the Algebraic Representation of the CAST Rule

Let's look at this diagram from the website Math is Fun (link) with the graphs of cosΘ, sinΘ, and tanΘ on top of each other on the interval of [0, 360 deg]:


We can clearly see which graphs are above the x-axis (and are hence positive) on the sub-intervals of quadrant 1, quadrant 2, quadrant 3, and quadrant 4.

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