Lesson Video: Signs of Trigonometric Functions in Quadrants Mathematics

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

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Video Transcript

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

First, letโ€™s consider a coordinate grid with an ๐‘ฅ- and ๐‘ฆ-axis. The top-right quadrant is labeled quadrant one. The top-left quadrant is quadrant two. The bottom-left quadrant is quadrant three. And the bottom-right quadrant is quadrant four. We know to the right of the origin, the ๐‘ฅ-values are positive. And to the left of the origin, the ๐‘ฅ-values are negative. In a similar way, above the origin, the ๐‘ฆ-values are positive. And below the origin, the ๐‘ฆ-values are negative. Letโ€™s add four points to our grid: the point ๐‘ฅ, ๐‘ฆ; the point negative ๐‘ฅ, ๐‘ฆ; the point negative ๐‘ฅ, negative ๐‘ฆ; and the point ๐‘ฅ, negative ๐‘ฆ.

And then in the first quadrant, we draw a line from the origin to the point ๐‘ฅ, ๐‘ฆ. And we let the angle created between the ๐‘ฅ-axis and this line be ๐œƒ. If we draw a vertical line from ๐‘ฅ, ๐‘ฆ to the ๐‘ฅ-axis, we see that weโ€™ve created a right-angled triangle with a horizontal distance from the origin of ๐‘ฅ and a vertical distance of ๐‘ฆ. And if weโ€™re given that itโ€™s one unit from the origin to the point ๐‘ฅ, ๐‘ฆ, we can use our trig functions to find out some things about this triangle.

For our three main trig functions, sine, cosine, and tangent, the sin of angle ๐œƒ will be equal to the opposite side length over the hypotenuse. The cos of angle ๐œƒ will be equal to the adjacent side length over the hypotenuse. And the tan of angle ๐œƒ will be the opposite side length over the adjacent side length. If we want to find sin of ๐œƒ, we can say that itโ€™s equal to ๐‘ฆ over one, since ๐‘ฆ is the opposite side length and the hypotenuse is one. Similarly, the cosine will be equal to ๐‘ฅ over one, the adjacent side length over the hypotenuse. And the tan of ๐œƒ will be equal to ๐‘ฆ over ๐‘ฅ.

We can simplify the sine and cosine to be ๐‘ฆ and ๐‘ฅ, respectively. And because we know that in the first quadrant all the ๐‘ฆ-values are positive, we can say that for angles falling in quadrant one, the sine value will be positive. Similarly, when we have ๐‘ฅ-values in the first quadrant, we know that the cosine value will also be positive. And tangent in the first quadrant will be a positive number over a positive number, which will also be positive.

Letโ€™s see how that changes if we move to the second quadrant. The distance from the origin to negative ๐‘ฅ, ๐‘ฆ is still one. Itโ€™s the opposite over the hypotenuse, ๐‘ฆ over one. But the cosine would then be negative ๐‘ฅ over one. In the second quadrant, weโ€™re dealing with negative ๐‘ฅ-values, which makes tan of ๐œƒ ๐‘ฆ over negative ๐‘ฅ. This means, in the second quadrant, the sine relationship remains positive. But the cosine relationship and the tangent relationship will be negative.

Leaving down to quadrant three, where weโ€™re dealing with negative ๐‘ฅ-coordinates and negative ๐‘ฆ-coordinates, sin of ๐œƒ will be negative ๐‘ฆ over one. cos ๐œƒ is negative ๐‘ฅ over one. We can simplify that to negative ๐‘ฆ and negative ๐‘ฅ. But something interesting happens with tangent. Itโ€™s equal to negative ๐‘ฆ over negative ๐‘ฅ, which simplifies to ๐‘ฆ over ๐‘ฅ. In the third quadrant, the tangent relationship is still positive. But in this quadrant, the sine and cosine relationships will be negative.

And now into the fourth quadrant, where the ๐‘ฅ-coordinate is positive and the ๐‘ฆ-coordinate is negative, sin of ๐œƒ is negative ๐‘ฆ over one. But cos of ๐œƒ is positive ๐‘ฅ over one, which gives us a negative sine and a positive cosine. And that will make our tangent negative ๐‘ฆ over ๐‘ฅ. The only positive relationship in the fourth quadrant is cosine. The negative ๐‘ฆ-values make the sine and tangent relationship negative.

What we discovered for each of these quadrants will be true for any angle that falls within that quadrant. Any angle in quadrant one will have positive sine, cosine, and tangent values. Any angle falling in quadrant two will only have a positive sine relationship. Angles in quadrant three will have positive tangent relationships. And angles in quadrant four will have positive cosine relationships.

There is a memory device we sometimes use to remember this. Itโ€™s called the CAST diagram, and it looks like this. In the CAST diagram, we indicate which trig relationships are positive in each quadrant. In the fourth quadrant, in the bottom right, cosine is positive, and sine and tangent are negative. In the first quadrant, in the top right, we have an A because all three relationships are positive. In the second quadrant, in the top left, sine is positive, with a negative cosine and a negative tangent. And in the third quadrant, the bottom left, tangent is positive, and sine and cosine are both negative.

Thereโ€™s one final thing we need to review before we look at some examples. And that is how we measure angles on a coordinate grid. The ๐‘ฅ-axis going in the right direction is called the initial side. And the terminal side is where the angle stops. If weโ€™re measuring from the initial side to the terminal side clockwise, weโ€™re measuring a positive angle measure. If weโ€™re measuring from the initial side to the terminal side in a clockwise manner, we will be measuring a negative angle measure. And finally, beginning at the initial side is a measure of zero degrees. From the initial side to the ๐‘ฆ-axis is 90 degrees, to the other side of the ๐‘ฅ-axis is 180 degrees, 90 degrees more gets us to 270, and finally back around to 360 degrees.

Now weโ€™re ready to look at some examples.

In which quadrant does the angle 288 degrees lie?

When we think about the four quadrants of the coordinate grid and label them one through four, we know that the initial side measures zero degrees. And then each additional quadrant is 90 more degrees. And then a full rotation is 360. When we measure angles in coordinate grids, we begin at the ๐‘ฅ-axis and proceed in a counterclockwise measure if weโ€™re dealing with a positive angle. And for us, that means weโ€™ll go from the initial side, just past 270, since we know that 288 falls between 270 and 360. And we see that this angle is in the fourth quadrant.

In our next example, weโ€™ll be given information about the sine and cosine of an angle and asked to find which quadrant it would lie in.

Determine the quadrant in which ๐œƒ lies if cos of ๐œƒ is greater than zero and sin of ๐œƒ is less than zero.

Weโ€™re trying to consider a coordinate grid and find which quadrant an angle would fall in. Weโ€™re told that cos of ๐œƒ is greater than zero, this means it has a positive cosine value, while the sin of ๐œƒ is less than zero, which means the sine has a negative value. One method we use for identifying the sine and cosine values in different quadrants is the CAST diagram that looks like this.

In the CAST diagram, we know that in the first quadrant, all values are positive. In the second quadrant, only the sine value is positive. In the third quadrant, only the tangent value is positive. And in the fourth quadrant, only cosine is positive. If we have a negative sine value and a positive cosine value, we can eliminate quadrant one as all values must be positive there. We can eliminate quadrant two as sine is positive there. In quadrant three, sine is negative, but so is cosine. And that means quadrant three will not work. In quadrant four, cosine is positive and sine is negative. And that means our angle ๐œƒ under these conditions must fall in the fourth quadrant.

Letโ€™s consider another example.

In which quadrant does ๐œƒ lie if sin of ๐œƒ equals one over the square root of two and cos of ๐œƒ equals one over the square root of two?

When we think about sine and cosine relationships, we know that sin of ๐œƒ is the opposite over the hypotenuse, while the cos of ๐œƒ is the adjacent side over the hypotenuse. But how do we translate that information into a coordinate grid?

In a coordinate grid, the sine, cosine, and tangent relationships will have either positive or negative values. And we can remember where each of these relationships will have positive values with the CAST diagram that looks like this. In quadrant one, all three trig relationships are positive. In quadrant two, only the sine relationship is positive. In quadrant three, only the tangent relationship is positive. And in quadrant four, only the cosine relationship is positive.

In this case, weโ€™re dealing with a positive sine relationship and a positive cosine relationship. We could also use the information weโ€™re given to find the tangent relationship, which would equal the opposite over the adjacent. For this angle, that would be one over one. And what weโ€™re seeing is that all three of these relationships are positive for this angle. And that means we must say it falls in the first quadrant.

In our next example, weโ€™ll consider an angle thatโ€™s larger than 360 degrees.

Is cos of 400 degrees positive or negative?

To answer this question, we need to figure out where 400 degrees would fall on a coordinate grid. If we label our standard coordinate grid from zero to 360 degrees, we need to think about what we would do with 400 degrees. Traveling counterclockwise one full rotation, weโ€™ve gone 360 degrees. But in order to get to 400, weโ€™ll need to go an additional 40 degrees, since 400 minus 360 equals 40. And that means the angle 400 would fall at the same place that the angle 40 degrees falls, here.

Now weโ€™ve identified where the angle 400 degrees would be on the coordinate grid, we need to think about how we would know if this is positive or negative. And to do that, we can use our CAST diagram that looks like this. Our CAST diagram tells us where trig relationships are positive in a coordinate grid. In the first quadrant, all three relationships are positive. In the second quadrant, only sine is positive. In the third quadrant, only tangent is positive. And in the fourth quadrant, only cosine is positive. Our angle falls in the first quadrant. In the first quadrant, sine, cosine, and tangent are positive. And that means the cos of 400 degrees will be positive.

Before we finish, letโ€™s review our key points. We can identify whether sine, cosine, and tangent will be positive or negative based on the quadrant in which their angle lies. In quadrant one, the sine, cosine, and tangent relationships will all be positive. For angles falling in quadrant two, the sine relationship will be positive, but the cosine and tangent relationships will be negative. For angles falling in quadrant three, the sine and cosine relationships will be negative, but the tangent relationship will be positive. And finally, in quadrant four, the sine relationship is negative, the cosine relationship is positive, and the tangent relationship is also negative. We often use the CAST diagram to remember this. These letters help us identify which values will be positive in which quadrant.

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