Lesson Explainer: Signs of Trigonometric Functions in Quadrants Mathematics

In this explainer, we will learn how to identify in which quadrant an angle lies and whether its sine, cosine, and tangent are positive or negative.

In order to determine the sign of the sine and cosine functions for a given angle, we will begin by recalling exactly what we mean by the sine and cosine functions of a given angle. To do this, we first need to recall exactly what we mean by the sine and cosine of an angle. The sine and cosine of an angle can be determined by using points on the unit circle.

The π‘₯-coordinate of this point will be cosπœƒ, where πœƒ is the angle measured in a counterclockwise direction from the positive π‘₯-axis. Similarly, the 𝑦-coordinate of this point will be sinπœƒ. When our angle πœƒ is acute, we have a right triangle where the sine and cosine functions are ratios of the lengths of the sides of this right triangle. However, this method allows us to define the sine and cosine of any angle from 0∘ to 360∘.

We can extend this even further by allowing directed angles. In this case, an angle measured in a clockwise direction is given by a negative number. We can also extend this further by allowing angles representing more than a full turn.

For example, we could represent the angle βˆ’120∘ as follows.

The measure of this angle is 120∘ and its direction is clockwise.

Therefore, to find the sine and cosine of a given angle, πœƒ, we can measure the angle πœƒ from the positive π‘₯-axis and then find the coordinates of the corresponding point on the unit circle.

Let’s see an example of how to apply this to an angle of 150∘. We start by measuring the angle 150∘ in the counterclockwise direction from the positive π‘₯-axis, on a diagram with the unit circle centered at the origin.

The coordinates of the point of intersection between the circle and line segment tell us the sine and cosine of the angle 150∘. We can see in the diagram that the 𝑦-coordinate of this point is positive; hence sin150∘ is also positive. Similarly, the π‘₯-coordinate of this point is negative; hence cos150∘ is also negative.

Let’s see an example of how we can use this to find the sign of the cosine of an angle larger than 360∘.

Example 1: Identifying the Sign of a Trigonometric Function of a Given Angle

Is cos400∘ positive or negative?

Answer

There are many different methods we can use to determine the sign of the cosine of a given angle. For example, we could use various trigonometric identities to evaluate this expression. However, we are going to focus on the graphical interpretation.

To do this, we first recall that the cosine of any angle is equal to the π‘₯-coordinate of the point 𝑃 on the unit circle, where 𝑂𝑃 makes an angle of πœƒ with the positive π‘₯-axis.

Therefore, we need to find the point 𝑃 such that 𝑂𝑃 makes an angle of 400∘ with the positive π‘₯-axis. Since 400=360+40∘∘∘, we get the following.

We can see that coscos400=40∘∘, since these two angles represent the same point on the unit circle. The cosine of this angle will be equal to the π‘₯-coordinate of 𝑃.

We can see in the diagram that the π‘₯-coordinate of 𝑃 is positive, since it lies to the right of the origin.

Hence, cos400∘ is positive.

In the previous example, we saw that the sign of the cosine of an angle was determined entirely by the value of the π‘₯-coordinate of the point where the terminal side of the angle intersects the unit circle. We can use this to determine the sign of the sine and cosine functions for any angle.

When the angle corresponds to a point on the unit circle to the right of the origin, then its π‘₯-coordinate is positive and hence its cosine is also positive. Similarly, when it corresponds to a point to the left of the origin, its π‘₯-coordinate is negative and hence its cosine must also be negative. We can also do this with the 𝑦-coordinates.

When the angle corresponds to a point on the unit circle that lies above the origin, then its 𝑦-coordinate is positive and hence its sine must also be positive. Similarly, when it corresponds to a point below the origin, its 𝑦-coordinate is negative and hence its sine must also be negative.

It would be useful to have terminology to talk about where in the coordinate plane a point and also an angle lie relative to the axes. Since the plane is naturally split into four sections by the axes, we can do this by labeling these sections we will call quadrants. We will start with the top right quadrant and number them counterclockwise.

We usually represent these with roman numerals and it is worth emphasizing that the axes themselves are not part of any quadrant. This allows us to describe where in the coordinate plane a point that does not sit on the axes lies.

For example, (1,2) lies in the first quadrant, (βˆ’3,2) lies in the second quadrant, (βˆ’3,βˆ’3) lies in the third quadrant, and (1,βˆ’2) lies in the fourth quadrant.

Definition: Quadrants in the Coordinate Planeβ€”Points

For any positive real numbers π‘₯ and 𝑦, we say that

  • (π‘₯,𝑦) lies in the first quadrant;
  • (βˆ’π‘₯,𝑦) lies in the second quadrant;
  • (βˆ’π‘₯,βˆ’π‘¦) lies in the third quadrant;
  • (π‘₯,βˆ’π‘¦) lies in the fourth quadrant.

Points on the axes are not said to lie in any quadrant.

In the same way, we can say that certain angles measured from the positive π‘₯-axis lie in a specific quadrant.

Definition: Quadrants in the Coordinate Planeβ€”Angles

Given an angle πœƒ in standard position, we say that πœƒ lies in the same quadrant as its terminal side.

Let’s see an example of how to determine the quadrant in which a given angle will lie.

Example 2: Determining in Which Quadrant a Given Angle Lies

In which quadrant does the angle 288∘ lie?

Answer

In order to determine the quadrant in which an angle will lie, we recall that quadrants are the 4 sections of the coordinate plane separated by the axes, and the quadrant of an angle is determined by the quadrant its terminal side lies in, if we measure from the positive π‘₯-axis.

We can do this in a sketch, where we remember that 288∘ is positive, so our angle is measured in a counterclockwise direction and we number the quadrants from the top right also in a counterclockwise direction.

Since 288=270+18∘∘∘, we can see that the terminal side will lie in the fourth quadrant. Hence, we can also say that 288∘ lies in the fourth quadrant.

We know the sign of the sine and cosine of an angle is determined entirely by its position on the coordinate plane. In fact, we can determine the sign solely by considering the quadrant in which the angle lies.

First, when our angle lies in the first or fourth quadrant, the π‘₯-coordinates of all points on its terminal side are positive. In particular, this means the π‘₯-coordinate of the point of intersection between the terminal side and the unit circle is also positive. Hence, the cosine of this angle is positive. Similarly, if the angle lies in the second or third quadrant, its cosine is negative.

Second, when our angle lies in the first or second quadrant, the 𝑦-coordinates of all points on its terminal side are positive. In particular, this means the 𝑦-coordinate of the intersection between the terminal side and the unit circle is also positive. Hence, the sine of this angle is positive. Similarly, if the angle lies in the third or fourth quadrant, its sine is negative.

We can summarize this as follows.

Definition: Signs of Trigonometric Functions in Different Quadrants

  • cosπœƒ>0, when πœƒ is in the first or fourth quadrant;
  • cosπœƒ<0, when πœƒ is in the second or third quadrant;
  • sinπœƒ>0, when πœƒ is in the first or second quadrant;
  • sinπœƒ<0, when πœƒ is in the third or fourth quadrant.

We can use these to find the sign of the tangent function in different quadrants. Recall the following trigonometric identity: tansincosπœƒβ‰‘πœƒπœƒ.

Since the tangent function is the quotient of the sine and cosine functions, its sign is determined by the sign of these functions. We do not need to worry about the cases where cosπœƒ=0, since these will occur on the axes themselves and not in a quadrant.

When πœƒ is in the first quadrant, sine and cosine are positive; therefore, tanπœƒ is the quotient of two positive numbers. Hence, tanπœƒ>0.

When πœƒ is in the second quadrant, sine is positive and cosine is negative; therefore, tanπœƒ is the quotient of a positive number and a negative one. Hence, tanπœƒ<0.

When πœƒ is in the third quadrant, sine is negative and cosine is negative; therefore, tanπœƒ is the quotient of two negative numbers. Hence, tanπœƒ>0.

Finally, when πœƒ is in the fourth quadrant, sine is negative and cosine is positive; therefore, tanπœƒ is the quotient of a negative number and a positive number. Hence, tanπœƒ<0.

This gives us the following.

Definition: Signs of the Tangent Function in Different Quadrants

  • tanπœƒ>0, when πœƒ is in the first or third quadrant;
  • tanπœƒ<0, when πœƒ is in the second or fourth quadrant.

We can determine the same information for the reciprocal trigonometric functions. However, since taking the reciprocal does not change the sign, the reciprocal trigonometric functions will have the same sign as their corresponding trigonometric functions.

There is a useful diagram, called the CAST diagram, that can be used to recall the signs of these trigonometric functions in different quadrants. We find this by labeling each quadrant with the trigonometric functions that are positive for arguments that lie there.

By using the acronym CAST to label the quadrants in a counterclockwise direction, we can quickly recall that all trigonometric functions are positive in the first quadrant, only the sine function is positive in the second quadrant, only the tangent function is positive in the third quadrant, and, finally, only the cosine function is positive in the fourth quadrant.

Let’s see an example of how we can use this diagram to determine the quadrant an angle must lie in, given information about the sine and cosine of the angle.

Example 3: Identifying Which Quadrant an Angle Lies in given Two of Its Trigonometric Ratios

In which quadrant does πœƒ lie if sinπœƒ=1√2 and cosπœƒ=1√2?

Answer

We want to determine the quadrant an angle lies in given information about the sine and cosine of this angle. To do this, we recall that quadrants are the 4 sections of the coordinate plane separated by the axes and the quadrant of an angle is determined by the quadrant its terminal side lies in, measured from the positive π‘₯-axis.

We could do this by using the definitions of the sine and cosine functions. However, a simpler method involves using the CAST diagram. We label the quadrants of the coordinate plane using the letters in the acronym, beginning in the fourth quadrant and moving in a counterclockwise direction.

The letters then tell us the quadrants for which the output of each trigonometric function is positive, where A means all, and C, S, and T mean cosine, sine, and tangent respectively. We are told in the question that sinπœƒ and cosπœƒ are both equal to 1√2. In other words, both the sine and cosine functions are positive. This only occurs in the first quadrant. Therefore, πœƒ is in the first quadrant.

The answer is: The first.

In our next example, we will determine the quadrant an angle lies in given the sign of the cosine and sine functions.

Example 4: Determining the Quadrant in Which an Angle Lies given Two of Its Trigonometric Ratios

Determine the quadrant in which πœƒ lies if cosπœƒ>0 and sinπœƒ<0.

Answer

In order to determine the quadrant in which a given angle will lie, we could use the definitions of the sine and cosine function. However, the CAST diagram can save us some time.

The CAST diagram tells us the quadrants in which the output of each trigonometric function is positive. We need to determine where cosine is positive and where sine is negative.

Let’s start with cosine being positive. The CAST diagram tells us cosine is positive for angles in the β€œC” or β€œA” quadrants, that is, the fourth and first quadrants respectively. Similarly, the CAST diagram tells us that, since sine is positive in the β€œA” and β€œS” quadrants, it will be negative in the other quadrants. In other words, the sine of an angle is negative for any angle in the third or fourth quadrant.

Since we need both of these to be true, πœƒ must lie in the fourth quadrant.

In our final example, we will see how to apply this to determine the sign of a reciprocal trigonometric function at a given angle.

Example 5: Determining the Sign of a Reciprocal Trigonometric Function of a Given Negative Angle

Is csc(βˆ’225)∘ positive or negative?

Answer

There are a number of methods we could use to determine the sign of the cosecant function of a given angle. For example, we could use various trigonometric identities or do this directly from the definition of the sine function. However, we will instead focus on the graphical interpretation by using the CAST diagram.

The CAST diagram tells us the quadrants for which various trigonometric functions are positive. To use this diagram, we first need to write csc(βˆ’225)∘, in terms of sine and cosine. We recall the following reciprocal trigonometric identity: cscsinπœƒβ‰‘1πœƒ; therefore, cscsin(βˆ’225)=1(βˆ’225).∘∘

This means we can determine the sign of csc(βˆ’225)∘ from the sign of sin(βˆ’225)∘. To do this, we need to determine which quadrant βˆ’225∘ lies in and we can achieve this by sketching the angle onto our CAST diagram. Remember, since this angle is negative, we need to measure the angle in a clockwise direction.

Since βˆ’225=βˆ’180βˆ’45∘∘∘, we see that this angle lies in the second quadrant. The CAST diagram tells us that the sine function is positive in the second quadrant. Therefore, sin(βˆ’225)>0.∘

In particular, this means 1(βˆ’225)sin∘ is the quotient of two positive numbers. Hence, csc(βˆ’225)∘ is positive.

Let’s finish by recapping some of the important points of this explainer.

Key Points

  • We can split the coordinate plane into four quadrants by using the axes. We number these 1–4, starting from the top right and moving in a counterclockwise direction.
  • An angle lying in a quadrant means its terminal side when measured from the positive π‘₯-axis lies in that quadrant.
  • We can determine the sign of all of the trigonometric functions by considering the quadrant the argument lies in.
  • We can identify the sign of sinπœƒ, cosπœƒ, and tanπœƒ for an angle πœƒ in each quadrant using the CAST diagram.
  • The reciprocal trigonometric functions will have the same sign as their corresponding trigonometric functions.

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