4 All Students Take Calculus Examples for Trigonometry

One of the most important skills that you need to develop as a trigonometry student is the ability to identify which trig functions are positive in each quadrant of the unit circle.

In my teaching experience, this can be a challenging concept for high school students who are just beginning their journey with trigonometric functions. 

Thankfully, the mnemonic device “All Students Take Calculus” provides an easy way to remember which trig ratios are positive in each quadrant.

And the good news is, you don’t have to actually take any calculus courses to understand how to apply this mnemonic device! Let’s get into what this strategy is and how to use it so that you can start solving trigonometry problems efficiently.

What Does “All Students Take Calculus” Mean?

All Students Take Calculus is a mnemonic device. If you have never used one before, a mnemonic device is a phrase that makes it easier to remember information that might otherwise be considered difficult to remember.

There are many different variations of this mnemonic device, such as:

• all students take calculus 
• all scientists teach chemistry
• all ships travel clockwise
• add sugar to coffee

Regardless of which one you choose, it is important to note that they all use the letters ASTC in that order.

Let’s stick with All Students Take Calculus and break down how this mnemonic device works to help students remember which trigonometric ratios are positive in each quadrant of the unit circle.

• First quadrant (All): In the first quadrant, all trigonometric functions are positive. This means that sine, cosine, and tangent have positive values for any given angle.
• Second quadrant (Students): In the second quadrant, only the sine function is positive, while the cosine function and tangent function will yield a negative value.
• Third quadrant (Take): In this quadrant, only the tangent function remains positive, while sine and cosine both carry a negative sign.
• Fourth quadrant (Calculus): In the final quadrant, only the cosine function is positive, while the sine and tangent will produce negative results.

Notice that the first letter of the mnemonic device corresponds to the first letter of the trigonometric ratios that are positive. This can be used as a simple cheat sheet that you can refer to when doing your homework assignments in order to develop a deeper understanding of trigonometry! 

Understanding All Students Take Calculus

Take a look at the following image of the unit circle. You can see that each quadrant has been labelled with a letter from the mnemonic device.

Let’s take a deeper dive into each quadrant to explore how the mnemonic All Students Take Calculus applies to the behavior of trig functions across the unit circle.

First Quadrant: “All”

In the first quadrant, all three trigonometric functions (sine, cosine, and tangent) are positive. We can explain this by looking at the x- and y- coordinates of any point on the unit circle in this quadrant. Both the x- and y-coordinates will always be positive in this quadrant. Therefore, no matter what the given angle is, you can confidently expect all trig ratios to be positive in this quadrant!

Since all measurements are positive in this quadrant, we often use it to solve right triangle problems in the real-world. For example, you may apply the Pythagorean theorem here or determining missing angles or missing side lengths in a real-world structure involving positive values.

Second Quadrant: “Students”

In the second quadrant, only the sine function remains positive. This is because while the y-coordinates of points on the unit circle are still positive, the x-coordinates become negative. The cosine function takes on a negative value here because it uses the adjacent side (which will be negative). This is also the case for the tangent function, which uses the opposite side (which will also be negative). Tangent is also the ratio of sine to cosine, which further explains why it will be negative!

It’s important to remember that angles in the second quadrant can be viewed as mirror images of angles from the first quadrant. Although the sine values remain the same for these angles, the negative sign for cosine flips the overall sign of the tangent. This quadrant is where students begin to encounter more complex trig identities, such as the double-angle formulas, which often require careful consideration of the quadrant to ensure the correct sign is used in calculations.

Third Quadrant: “Take”

The third quadrant is where the tangent function is the only trig function that remains positive. In this quadrant, both the x- and y-coordinates of the points on the unit circle are negative, meaning that both the sine function and the cosine function take on negative values. However, since tangent is the ratio of sine to cosine, and dividing two negative numbers results in a positive value, the tangent function remains positive here.

Understanding the third quadrant becomes crucial when dealing with complex trigonometric problems that involve negative values, such as those that include the square roots of negative numbers. Additionally, more advanced trigonometry students will encounter the half-angle formula in this quadrant, which becomes especially useful in certain calculus courses. 

Fourth Quadrant: “Calculus”

In the fourth quadrant, only the cosine function is positive. This occurs because, while the x-coordinates of points on the unit circle remain positive, the y-coordinates become negative. As a result, the sine function and tangent function both take on negative values in this quadrant.

Mastering the fourth quadrant is especially important for students as they transition from high school trigonometry into more advanced calculus courses. In calculus, students will delve deeper into trig identities like the double-angle formulas and learn to apply trigonometry to solve real-world problems that involve rotations, wave motion, and other cyclic phenomena.

4 All Students Take Calculus Examples

In order to help you master this mnemonic device, I have put together 4 All Students Take Calculus examples that will help you practice predicting the sign of trigonometric ratios in each quadrant!

Example 1: Determining Positivity in the First Quadrant

In this first example, we will explore how to apply the mnemonic to the first quadrant of the unit circle.

Problem: Find the values of sine, cosine, and tangent for an angle of 30° in the first quadrant. Identify which of these trig functions will be positive.

Since 30° is in the first quadrant, we know from the mnemonic All Students Take Calculus that all three functions (sine, cosine, and tangent) are positive.

\[\sin(30^\circ) = \frac{1}{2}\]

\[\cos(30^\circ) = \frac{\sqrt{3}}{2}\]

\[\tan(30^\circ) = \frac{1}{\sqrt{3}}\]

Example 2: Applying the Mnemonic in the Second Quadrant

In this next example, let’s take a look at how we can apply our memory aid in the second quadrant of the unit circle!

Problem: You are given an angle of 120°. Determine the sign of sine, cosine, and tangent in the second quadrant.

Since 120° lies in the second quadrant, we use the mnemonic All Students Take Calculus to remember that only sine is positive.

\[\sin(120^\circ) = \frac{\sqrt{3}}{2} \quad \text{(Positive)}\]

\[\cos(120^\circ) = -\frac{1}{2} \quad \text{(Negative)}\]

\[\tan(120^\circ) = -\sqrt{3} \quad \text{(Negative)}\]

Example 3: Positivity in the Third Quadrant

Next, let’s explore an All Students Take Calculus example in the third quadrant of the unit circle.

Problem: Determine the sign of sine, cosine, and tangent for an angle of 210° in the third quadrant.

The angle 210° is in the third quadrant, where only tangent is positive, according to All Students Take Calculus.

\[\sin(210^\circ) = -\frac{1}{2} \quad \text{(Negative)}\]

\[\cos(210^\circ) = -\frac{\sqrt{3}}{2} \quad \text{(Negative)}\]

\[\tan(210^\circ) = \frac{1}{\sqrt{3}} \quad \text{(Positive)}\]

Example 4: Finding the Sign of Functions in the Fourth Quadrant

Consider this last example, which focuses on applying the memory aid in the fourth quadrant of the unit circle.

Problem: An angle measures 330°. Using the fourth quadrant, determine which trig functions will be positive or negative.

Since 330° lies in the fourth quadrant, we can use All Students Take Calculus to determine that only the cosine function is positive.

\[\sin(330^\circ) = -\frac{1}{2} \quad \text{(Negative)}\]

\[\cos(330^\circ) = \frac{\sqrt{3}}{2} \quad \text{(Positive)}\]

\[\tan(330^\circ) = -\frac{1}{\sqrt{3}} \quad \text{(Negative)}\]

Using All Students Take Calculus in Your Studies

In my 10+ years experience teaching trigonometry, I have seen that mastering this concept comes down to understanding the relationships between trigonometric functions and how they behave across the unit circle.

I have used the mnemonic device All Students Take Calculus to help thousands of students by providing them with an easy way to recall which trig ratios are positive in each quadrant. This really helps such students by simplifying very complex problems into much more manageable steps. 

By knowing that all trig functions are positive in the first quadrant, only sine is positive in the second quadrant, only the tangent function is positive in the third quadrant, and only the cosine function is positive in the fourth quadrant, you will find yourself able to tackle problems more confidently.

This understanding will be very important as you move through high school and begin to explore more advanced concepts, such as double-angle formulas, the half-angle formula, and the various trig identities encountered in more advanced calculus courses.

Whether you’re working with right triangle problems or solving for missing side lengths and missing angles, the ability to quickly determine the signs of trigonometric ratios in each quadrant will significantly improve your problem-solving skills.

As you continue practicing, be sure to use this memory aid as a cheat sheet in your homework assignments to reinforce your understanding of the unit circle and the behavior of trigonometric functions!

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