Trigonometric functions are often used in math (including calculus), and they also have real-life applications (such as signal processing). However, it can be tricky to remember the signs of these functions.
So, how do you remember trigonometric functions and their signs? Use the memory device ASTC (all students take calculus) to label quadrants 1, 2, 3, & 4. A means all trig functions are positive and S, T, C stand for sine, tangent, & cosine. Also, cosine is the x-coordinate of a point on the unit circle, and sine is the y-coordinate of a point on the unit circle.
Of course, once you know the signs of sine, cosine, and tangent in a given quadrant, you can also find the signs of their reciprocal trig functions (cosecant, secant, and cotangent).
In this article, we’ll talk about ways to remember the signs of trigonometric functions, depending on what quadrant they are in (and the related angles). We’ll also talk about some special cases (when the trig functions are zero or undefined).
Let’s get started.
Trigonometric Functions & Their Signs
There are two methods to help you remember the signs of trigonometric functions:
- By Quadrant – use the memory device ASTC (all students take calculus) to label the 4 quadrants and find out which functions are positive or negative (more detail on this below).
- By Coordinates – remember that the cosine of an angle gives us the x coordinate of the corresponding triangle, and the sine gives us the y coordinate of the corresponding triangle (more detail on this below).
Trigonometric Functions & Their Signs By Quadrants
We can use the memory device ASTC (all students take calculus) to keep track of the signs of trigonometric functions by quadrant. We assign each quadrant a letter as follows:
- A = 1st Quadrant (top right): All trig functions are positive.
- S = 2nd Quadrant (top left): Sine is positive (along with cosecant, the reciprocal of sine).
- T = 3rd Quadrant (bottom left): Tangent is positive (along with cotangent, the reciprocal of tangent).
- C = 4th Quadrant (bottom right): Cosine is positive (along with secant, the reciprocal of cosine).
You can see a graph with the quadrant labels below.
Notice how the memory device works: remember the phrase “all students take calculus” and take the first letter of each word: ASTC. Remember that A stands for all, S stands for sine, t stands for tangent, and c stands for cosine.
We put them in the same order as the number labels for the quadrants: 1, 2, 3, 4.
However, this isn’t the only way to remember and understand the signs of trig functions.
Trigonometric Functions & Their Signs By Coordinates
Another way to keep track of signs of trig functions is to use what we know about the coordinates of a point on the unit circle and the trig functions.
The image below shows a point P on the unit circle, the line segment between P and O (where O is the origin), the angle between the line segment PO and the x-axis, and the coordinates of the point P.
If the point P has coordinates (xP, yP) and the angle between OP and the x-axis is A, then we have:
- xP = cos(A)
- yP = sin(A)
In other words, the cosine of the angle A gives us the x coordinate of the point P, and the sine of the angle A gives us the y coordinate of the point P (note that this only works for points on the unit circle centered at the origin).
For a circle of radius R centered at the origin, the point P would have coordinates:
- xP = Rcos(A)
- yP = Rsin(A)
So, how does this help us with the signs of trig functions?
Well, we can think of a point P as being either to the left or to the right of the y-axis:
- If the point P is to the right of the y-axis, then xP is positive, and so cos(A) must be positive.
- If the point P is to the left of the y-axis, then xP is negative, and so cos(A) must be negative.
Similarly, we can think of a point P as being either above or below of the x-axis:
- If the point P is above the x-axis, then yP is positive, and so sin(A) must be positive.
- If the point P is below the x-axis, then yP is negative, and so sin(A) must be negative.
Once we know the signs of the sine and cosine functions, we can easily find the signs of the other trig functions:
- tan(A) = sin(A) / cos(A)
- sec(A) = 1 / cos(A)
- csc(A) = 1 / sin(A)
- cot(A) = 1 /tan(A)
So, for example, if sin(A) and cos(A) have the same sign, then tan(A) must be positive. If sin(A) and cos(A) have opposite signs, then tan(A) is negative.
Also:
- sec(A) has the same sign as cos(A)
- csc(A) has the same sign as sin(A)
- cot(A) has the same sign as tan(A)
Note that tangent, secant, cosecant, and cotangent can be undefined when they have zero denominators (this can happen when either sine or cosine are equal to zero – more on this later).
The table below summarizes the signs of tangent, secant, cosecant, and cotangent, based on the signs of sin(A) and cos(A).
| Function | Q1 | Q2 | Q3 | Q4 |
|---|---|---|---|---|
| Sine | + | + | – | – |
| Cosine | + | – | – | + |
| Tangent | + | – | + | – |
| Secant | + | – | – | + |
| Cosecant | + | + | – | – |
| Cotangent | + | – | + | – |
function by quadrant (Q1, Q2, Q3, Q4
correspond to A, S, T, C).
Sine
Let’s review when sine is positive, negative, zero, and undefined.
When Is Sine Positive?
Sine is positive in the 1st and 2nd quadrants, when the y-coordinate is above the x-axis. This occurs when the angle A is between 0 and π radians (0 and 180 degrees).
When Is Sine Negative?
Sine is negative in the 3rd and 4th quadrants, when the y-coordinate is below the x-axis. This occurs when the angle A is between π and 2π radians (180 and 360 degrees).
When Is Sine 1?
Sine is 1 when the angle A is π/2 radians (90 degrees). This means that the point on the unit circle lies on the y-axis (between the 1st and 2nd quadrants).
Sine will also be 1 when A is any angle that is equivalent to π/2, such as:
- -7π/2, -3π/2, π/2, 5π/2, 9π/2, … (4k + 1)π / 2 for any integer k
Note that sine is -1 when the angle A is –π/2 radians (-90 degrees) or equivalently 3π/2 radians (270 degrees). This means that the point on the unit circle lies on the y-axis (between the 3rd and 4th quadrants).
Sine will also be -1 when A is any angle that is equivalent to -π/2 radians, such as:
- -5π/2, -π/2, 3π/2, 7π/2, … (4k + 3)π / 2 for any integer k
When Is Sine 0?
Sine is 0 when the angle A is 0 radians (0 degrees). This means that the point on the unit circle is on the x-axis (between the 1st and 4th quadrants).
Sine is also 0 when the angle A is π radians (180 degrees). This means that the point on the unit circle lies on the x-axis (between the 2nd and 3rd quadrants).
Sine will also be 0 when A is any angle that is equivalent to 0 or π, such as:
- -3π, -2π, -π, 0, π, 2π, 3π … kπ for any integer k
When Is Sine Undefined?
Sine is never undefined. It can take on any value between -1 and 1, and we can use any angle as an input into the sine function.
When Is Sine Equal To Cosine?
Sine is equal to cosine when the angle A is π/4 radians (45 degrees). This means that the point on the unit circle is on the halfway point of the circumference between (0, 0) and (0, 1) in the first quadrant.
Sine is also equal to cosine when the angle A is any angle that is equivalent to π/4, such as:
- -15π/4, -7π/4, π/4, 9π/4, 17π/4, … (8k + 1)π / 4 for any integer k
Cosine
Let’s review when cosine is positive, negative, zero, and undefined.
When Is Cosine Positive?
Cosine is positive in the 1st and 4th quadrants, when the x-coordinate is to the right of the y-axis. This occurs when the angle A is between –π/2 and π/2 radians (-90 and 90 degrees).
When Is Cosine Negative?
Cosine is negative in the 2nd and 3rd quadrants, when the x-coordinate is to the left of the y-axis. This occurs when the angle A is between π/2 and 3π/2 radians (90 and 270 degrees).
When Is Cosine 1?
Cosine is 1 when the angle A is 0 radians (0 degrees). This means that the point on the unit circle lies on the x-axis (between the 1st and 4th quadrants).
Cosine will also be 1 when A is any angle that is equivalent to 0, such as:
- -4π, -2π, 0, 2π, 4π, … 2kπ for any integer k
Note that cosine is -1 when the angle A is π radians (180 degrees) or equivalently -π radians (-180 degrees). This means that the point on the unit circle lies on the x-axis (between the 2nd and 3rd quadrants).
Cosine will also be -1 when A is any angle that is equivalent to π radians, such as:
- -3π, -π, π, 3π … (2k + 1)π for any integer k
When Is Cosine 0?
Cosine is 0 when the angle A is π/2 radians (90 degrees). This means that the point on the unit circle is on the y-axis (between the 1st and 2nd quadrants).
Cosine is also 0 when the angle A is 3π/2 radians (270 degrees). This means that the point on the unit circle lies on the y-axis (between the 3rd and 4th quadrants).
Cosine will also be 0 when A is any angle that is equivalent to π/2 or 3π/2, such as:
- -3π/2, -π/2, π/2, 3π/2, … (2k + 1)π / 2 for any integer k
When Is Cosine Undefined?
Cosine is never undefined. It can take on any value between -1 and 1, and we can use any angle as an input into the cosine function.
Tangent
Let’s review when tangent is positive, negative, zero, and undefined.
When Is Tangent Positive?
Tangent is positive in the 1st and 3rd quadrants, when sine and cosine have the same sign (both positive or both negative). This occurs when the angle A is between 0 and π/2 radians (0 and 90 degrees) or between π and 3π/2 radians (180 and 270 degrees).
When Is Tangent Negative?
Tangent is negative in the 2nd and 4th quadrants, when sine and cosine have opposite signs (one positive, the other negative). This occurs when the angle A is between π/2 and π radians (90 and 180 degrees) or between 3π/2 and 2π radians (270 and 360 degrees).
When Is Tangent 1?
Tangent is 1 when sine and cosine have the same value (including sign). This happens when the angle A is π/4 radians (45 degrees), and sin(A) = cos(A) = √2 / 2.
This means that the point on the unit circle is on the halfway point of the circumference between (0, 0) and (0, 1) in the first quadrant.
Tangent will also be 1 when angle A is 5π/4 radians (225 degrees), and sin(A) = cos(A) = -√2 / 2.
This means that the point on the unit circle is on the halfway point of the circumference between (-1, 0) and (0, -1) in the third quadrant.
Tangent will also be 1 when A is any angle that is equivalent to π/4 or 5π/4, such as:
- -7π/4, -3π/4, π/4, 5π/4, 9π/4, … (4k + 1)π / 4 for any integer k
Note that tangent is -1 when sine and cosine have opposite values (one positive and one negative). This occurs when the angle A is 3π/4 radians (135 degrees).
This means that the point on the unit circle is on the halfway point of the circumference between (0, 1) and (-1, 0) in the second quadrant.
Tangent will also be -1 when angle A is 7π/4 radians (315 degrees).
This means that the point on the unit circle is on the halfway point of the circumference between (0, -1) and (1, 0) in the fourth quadrant.
Tangent will also be -1 when A is any angle that is equivalent to 3π/4 or 7π/4 radians, such as:
- -9π/4, -5π/4, -π/4, 3π/4, 7π/4, … (4k + 3)π / 4 for any integer k
When Is Tangent 0?
Tangent is zero whenever sine is zero. This is due to the fact that tan(A) = sin(A) / cos(A), which means sin(A) is the numerator of tan(A).
We know when sine is zero from our earlier discussion (see “When Is Sine Zero” above).
Tangent is 0 when the angle A is 0 radians (0 degrees). This means that the point on the unit circle is on the x-axis (between the 1st and 4th quadrants).
Tangent is also 0 when the angle A is π radians (180 degrees). This means that the point on the unit circle lies on the x-axis (between the 2nd and 3rd quadrants).
Tangent will also be 0 when A is any angle that is equivalent to 0 or π, such as:
- -3π, -2π, -π, 0, π, 2π, 3π … kπ for any integer k
When Is Tangent Undefined?
Tangent is undefined whenever cosine is zero. This is due to the fact that tan(A) = sin(A) / cos(A), which means cos(A) is the denominator of tan(A).
We know when cosine is zero from our earlier discussion (see “When Is Cosine Zero” above).
Tangent is 0 when the angle A is π/2 radians (90 degrees). This means that the point on the unit circle is on the y-axis (between the 1st and 2nd quadrants).
Tangent is also 0 when the angle A is 3π/2 radians (270 degrees). This means that the point on the unit circle lies on the y-axis (between the 3rd and 4th quadrants).
Tangent will also be 0 when A is any angle that is equivalent to π/2 or 3π/2, such as:
- -3π/2, -π/2, π/2, 3π/2, … (2k + 1)π / 2 for any integer k
Secant
Let’s review when secant is positive, negative, zero, and undefined.
When Is Secant Positive?
Secant is positive whenever cosine is positive. This is due to the fact that sec(A) = 1 / cos(A), which means sec(A) is the reciprocal of cos(A).
We know when cosine is positive from our earlier discussion (see “When Is Cosine Positive” above).
Secant is positive in the 1st and 4th quadrants, when the x-coordinate is to the right of the y-axis. This occurs when the angle A is between –π/2 and π/2 radians (-90 and 90 degrees).
When Is Secant Negative?
Secant is negative whenever cosine is negative. This is due to the fact that sec(A) = 1 / cos(A), which means sec(A) is the reciprocal of cos(A).
We know when cosine is negative from our earlier discussion (see “When Is Cosine Negative” above).
Secant is negative in the 2nd and 3rd quadrants, when the x-coordinate is to the left of the y-axis. This occurs when the angle A is between π/2 and 3π/2 radians (90 and 270 degrees).
When Is Secant 1?
Secant is 1 whenever cosine is 1. This is due to the fact that sec(A) = 1 / cos(A), which means sec(A) is the reciprocal of cos(A).
We know when cosine is 1 from our earlier discussion (see “When Is Cosine 1” above).
Secant is 1 when the angle A is 0 radians (0 degrees). This means that the point on the unit circle lies on the x-axis (between the 1st and 4th quadrants).
Secant will also be 1 when A is any angle that is equivalent to 0, such as:
- -4π, -2π, 0, 2π, 4π, … 2kπ for any integer k
Note that Secant is -1 when the angle A is π radians (180 degrees) or equivalently -π radians (-180 degrees). This means that the point on the unit circle lies on the x-axis (between the 2nd and 3rd quadrants).
Secant will also be -1 when A is any angle that is equivalent to π radians, such as:
- -3π, -π, π, 3π … (2k + 1)π for any integer k
When Is Secant 0?
Secant is never zero. This is due to the fact that sec(A) = 1 / cos(A), and cos(A) only takes on finite values (between -1 and 1).
When Is Secant Undefined?
Secant is undefined whenever cosine is zero. This is due to the fact that sec(A) = 1 / cos(A), and cos(A) = 0 would give us a zero denominator (undefined).
We know when cosine is zero from our earlier discussion (see “When Is Cosine Zero” above).
Secant is undefined when the angle A is π/2 radians (90 degrees). This means that the point on the unit circle is on the y-axis (between the 1st and 2nd quadrants).
Secant is also undefined when the angle A is 3π/2 radians (270 degrees). This means that the point on the unit circle lies on the y-axis (between the 3rd and 4th quadrants).
Secant will also be undefined when A is any angle that is equivalent to π/2 or 3π/2, such as:
- -3π/2, -π/2, π/2, 3π/2, … (2k + 1)π / 2 for any integer k
When Is Secant Equal To Cosecant?
Secant is equal to cosecant whenever cosine is equal to sine, since:
- sec(x) = csc(x)
- 1 / cos(x) = 1 / sin(x)
- sin(x) = cos(x)
We know when sine and cosine are equal from our earlier discussion (see “When Is Sine Equal to Cosine” above).
Secant is equal to cosecant when the angle A is π/4 radians (45 degrees). This means that the point on the unit circle is on the halfway point of the circumference between (0, 0) and (0, 1) in the first quadrant.
Secant is also equal to cosecant when the angle A is any angle that is equivalent to π/4, such as:
- -15π/4, -7π/4, π/4, 9π/4, 17π/4, … (8k + 1)π / 4 for any integer k
Cosecant
Let’s review when cosecant is positive, negative, zero, and undefined.
When Is Cosecant Positive?
Cosecant is positive whenever sine is positive. This is due to the fact that csc(A) = 1 / sin(A), which means csc(A) is the reciprocal of sin(A).
We know when sine is positive from our earlier discussion (see “When Is Sine Positive” above).
Cosecant is positive in the 1st and 2nd quadrants, when the y-coordinate is above the x-axis. This occurs when the angle A is between 0 and π radians (0 and 180 degrees).
When Is Cosecant Negative?
Cosecant is negative whenever sine is negative. This is due to the fact that csc(A) = 1 / sin(A), which means csc(A) is the reciprocal of sin(A).
We know when sine is negative from our earlier discussion (see “When Is Sine Negative” above).
Cosecant is negative in the 3rd and 4th quadrants, when the y-coordinate is below the x-axis. This occurs when the angle A is between π and 2π radians (180 and 360 degrees).
When Is Cosecant 1?
Cosecant is 1 whenever sine is 1. This is due to the fact that csc(A) = 1 / sin(A), which means csc(A) is the reciprocal of sin(A).
We know when sine is 1 from our earlier discussion (see “When Is Sine 1” above).
Cosecant is 1 when the angle A is π/2 radians (90 degrees). This means that the point on the unit circle lies on the y-axis (between the 1st and 2nd quadrants).
Cosecant will also be 1 when A is any angle that is equivalent to π/2, such as:
- -7π/2, -3π/2, π/2, 5π/2, 9π/2, … (4k + 1)π / 2 for any integer k
Note that Cosecant is -1 when the angle A is –π/2 radians (-90 degrees) or equivalently 3π/2 radians (270 degrees). This means that the point on the unit circle lies on the y-axis (between the 3rd and 4th quadrants).
Cosecant will also be -1 when A is any angle that is equivalent to -π/2 radians, such as:
- -5π/2, -π/2, 3π/2, 7π/2, … (4k + 3)π / 2 for any integer k
When Is Cosecant 0?
Cosecant is never zero. This is due to the fact that csc(A) = 1 / sin(A), and csc(A) only takes on finite values (between -1 and 1).
When Is Cosecant Undefined?
Cosecant is undefined whenever sine is zero. This is due to the fact that csc(A) = 1 / sin(A), and sin(A) = 0 would give us a zero denominator (undefined).
We know when sine is zero from our earlier discussion (see “When Is Sine Zero” above).
Cosecant is 0 when the angle A is 0 radians (0 degrees). This means that the point on the unit circle is on the x-axis (between the 1st and 4th quadrants).
Cosecant is also 0 when the angle A is π radians (180 degrees). This means that the point on the unit circle lies on the x-axis (between the 2nd and 3rd quadrants).
Cosecant will also be 0 when A is any angle that is equivalent to 0 or π, such as:
- -3π, -2π, -π, 0, π, 2π, 3π … kπ for any integer k
Cotangent
Let’s review when cotangent is positive, negative, zero, and undefined.
When Is Cotangent Positive?
Cotangent is positive whenever tangent is negative. This is due to the fact that cot(A) = 1 / tan(A), which means cot(A) is the reciprocal of tan(A).
We know when tangent is positive from our earlier discussion (see “When Is Tangent Positive” above).
Cotangent is positive in the 1st and 3rd quadrants, when sine and cosine have the same sign (both positive or both negative). This occurs when the angle A is between 0 and π/2 radians (0 and 90 degrees) or between π and 3π/2 radians (180 and 270 degrees).
When Is Cotangent Negative?
Cotangent is negative whenever tangent is negative. This is due to the fact that cot(A) = 1 / tan(A), which means cot(A) is the reciprocal of tan(A).
We know when tangent is negative from our earlier discussion (see “When Is Tangent Negative” above).
Cotangent is negative in the 2nd and 4th quadrants, when sine and cosine have opposite signs (one positive, the other negative). This occurs when the angle A is between π/2 and π radians (90 and 180 degrees) or between 3π/2 and 2π radians (270 and 360 degrees).
When Is Cotangent 1?
Cotangent is 1 whenever tangent is 1. This is due to the fact that cot(A) = 1 / tan(A), which means cot(A) is the reciprocal of tan(A).
We know when tangent is 1 from our earlier discussion (see “When Is Tangent 1” above).
Cotangent is 1 when sine and cosine have the same value (including sign). This happens when the angle A is π/4 radians (45 degrees), and sin(A) = cos(A) = √2 / 2.
This means that the point on the unit circle is on the halfway point of the circumference between (0, 0) and (0, 1) in the first quadrant.
Cotangent will also be 1 when angle A is 5π/4 radians (225 degrees), and sin(A) = cos(A) = -√2 / 2.
This means that the point on the unit circle is on the halfway point of the circumference between (-1, 0) and (0, -1) in the third quadrant.
Cotangent will also be 1 when A is any angle that is equivalent to π/4 or 5π/4, such as:
- -7π/4, -3π/4, π/4, 5π/4, 9π/4, … (4k + 1)π / 4 for any integer k
Note than cotangent will be -1 whenever tangent is -1.
When Is Cotangent 0?
Cotangent is zero whenever cosine is zero. This is due to the fact that cot(A) = cos(A) / sin(A), which means cos(A) is the numerator of tan(A).
We know when cosine is zero from our earlier discussion (see “When Is Cosine Zero” above).
Cotangent is 0 when the angle A is π/2 radians (90 degrees). This means that the point on the unit circle is on the y-axis (between the 1st and 2nd quadrants).
Cotangent is also 0 when the angle A is 3π/2 radians (270 degrees). This means that the point on the unit circle lies on the y-axis (between the 3rd and 4th quadrants).
Cotangent will also be 0 when A is any angle that is equivalent to π/2 or 3π/2, such as:
- -3π/2, -π/2, π/2, 3π/2, … (2k + 1)π / 2 for any integer k
When Is Cotangent Undefined?
Cotangent is undefined whenever sine is zero. This is due to the fact that cot(A) = cos(A) / sin(A), which means sin(A) is the denominator of tan(A).
We know when sine is zero from our earlier discussion (see “When Is Sine Zero” above).
Cotangent is 0 when the angle A is 0 radians (0 degrees). This means that the point on the unit circle is on the x-axis (between the 1st and 4th quadrants).
Cotangent is also 0 when the angle A is π radians (180 degrees). This means that the point on the unit circle lies on the x-axis (between the 2nd and 3rd quadrants).
Cotangent will also be 0 when A is any angle that is equivalent to 0 or π, such as:
- -3π, -2π, -π, 0, π, 2π, 3π … kπ for any integer k
Conclusion
Now you know all about the signs of trigonometric functions and how to remember them (drawing out coordinate axes with numbers and letters ASTC helps!)
You can learn about the output values of sine here.
You can learn about the output values of cosine here.
You can learn about inverse trigonometric functions here.
You can learn more about real life examples of how trig functions are used in my article here.
You can learn about the Law of Sines (and how it is used to solve triangles) in this article.
You can learn more about the Law of Cosines (and how it helps you to solve a triangle) here.
You can learn about other nonlinear functions in my article here.
I hope you found this article helpful. If so, please share it with someone who can use the information.
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~Jonathon
