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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.
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First, letβs consider a coordinate grid with an π₯- and π¦-axis.
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The top-right quadrant is labeled quadrant one.
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The top-left quadrant is quadrant two.
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The bottom-left quadrant is quadrant three.
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And the bottom-right quadrant is quadrant four.
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We know to the right of the origin, the π₯-values are positive.
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And to the left of the origin, the π₯-values are negative.
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In a similar way, above the origin, the π¦-values are positive.
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And below the origin, the π¦-values are negative.
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Letβs add four points to our grid: the point π₯, π¦; the point negative π₯, π¦; the point negative π₯, negative π¦; and the point π₯, negative π¦.
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And then in the first quadrant, we draw a line from the origin to the point π₯, π¦.
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And we let the angle created between the π₯-axis and this line be π.
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If we draw a vertical line from π₯, π¦ to the π₯-axis, we see that weβve created a right-angled triangle with a horizontal distance from the origin of π₯ and a vertical distance of π¦.
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And if weβre given that itβs one unit from the origin to the point π₯, π¦, we can use our trig functions to find out some things about this triangle.
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For our three main trig functions, sine, cosine, and tangent, the sin of angle π will be equal to the opposite side length over the hypotenuse.
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The cos of angle π will be equal to the adjacent side length over the hypotenuse.
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And the tan of angle π will be the opposite side length over the adjacent side length.
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If we want to find sin of π, we can say that itβs equal to π¦ over one, since π¦ is the opposite side length and the hypotenuse is one.
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Similarly, the cosine will be equal to π₯ over one, the adjacent side length over the hypotenuse.
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And the tan of π will be equal to π¦ over π₯.
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We can simplify the sine and cosine to be π¦ and π₯, respectively.
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And because we know that in the first quadrant all the π¦-values are positive, we can say that for angles falling in quadrant one, the sine value will be positive.
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Similarly, when we have π₯-values in the first quadrant, we know that the cosine value will also be positive.
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And tangent in the first quadrant will be a positive number over a positive number, which will also be positive.
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Letβs see how that changes if we move to the second quadrant.
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The distance from the origin to negative π₯, π¦ is still one.
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Itβs the opposite over the hypotenuse, π¦ over one.
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But the cosine would then be negative π₯ over one.
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In the second quadrant, weβre dealing with negative π₯-values, which makes tan of π π¦ over negative π₯.
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This means, in the second quadrant, the sine relationship remains positive.
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But the cosine relationship and the tangent relationship will be negative.
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Leaving down to quadrant three, where weβre dealing with negative π₯-coordinates and negative π¦-coordinates, sin of π will be negative π¦ over one.
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cos π is negative π₯ over one.
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We can simplify that to negative π¦ and negative π₯.
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But something interesting happens with tangent.
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Itβs equal to negative π¦ over negative π₯, which simplifies to π¦ over π₯.
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In the third quadrant, the tangent relationship is still positive.
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But in this quadrant, the sine and cosine relationships will be negative.
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And now into the fourth quadrant, where the π₯-coordinate is positive and the π¦-coordinate is negative, sin of π is negative π¦ over one.
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But cos of π is positive π₯ over one, which gives us a negative sine and a positive cosine.
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And that will make our tangent negative π¦ over π₯.
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The only positive relationship in the fourth quadrant is cosine.
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The negative π¦-values make the sine and tangent relationship negative.
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What we discovered for each of these quadrants will be true for any angle that falls within that quadrant.
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Any angle in quadrant one will have positive sine, cosine, and tangent values.
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Any angle falling in quadrant two will only have a positive sine relationship.
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Angles in quadrant three will have positive tangent relationships.
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And angles in quadrant four will have positive cosine relationships.
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There is a memory device we sometimes use to remember this.
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Itβs called the CAST diagram, and it looks like this.
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In the CAST diagram, we indicate which trig relationships are positive in each quadrant.
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In the fourth quadrant, in the bottom right, cosine is positive, and sine and tangent are negative.
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In the first quadrant, in the top right, we have an A because all three relationships are positive.
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In the second quadrant, in the top left, sine is positive, with a negative cosine and a negative tangent.
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And in the third quadrant, the bottom left, tangent is positive, and sine and cosine are both negative.
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Thereβs one final thing we need to review before we look at some examples.
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And that is how we measure angles on a coordinate grid.
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The π₯-axis going in the right direction is called the initial side.
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And the terminal side is where the angle stops.
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If weβre measuring from the initial side to the terminal side clockwise, weβre measuring a positive angle measure.
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If weβre measuring from the initial side to the terminal side in a clockwise manner, we will be measuring a negative angle measure.
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And finally, beginning at the initial side is a measure of zero degrees.
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From the initial side to the π¦-axis is 90 degrees, to the other side of the π₯-axis is 180 degrees, 90 degrees more gets us to 270, and finally back around to 360 degrees.
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Now weβre ready to look at some examples.
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In which quadrant does the angle 288 degrees lie?
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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.
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And then each additional quadrant is 90 more degrees.
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And then a full rotation is 360.
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When we measure angles in coordinate grids, we begin at the π₯-axis and proceed in a counterclockwise measure if weβre dealing with a positive angle.
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And for us, that means weβll go from the initial side, just past 270, since we know that 288 falls between 270 and 360.
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And we see that this angle is in the fourth quadrant.
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In our next example, weβll be given information about the sine and cosine of an angle and asked to find which quadrant it would lie in.
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Determine the quadrant in which π lies if cos of π is greater than zero and sin of π is less than zero.
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Weβre trying to consider a coordinate grid and find which quadrant an angle would fall in.
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Weβre told that cos of π is greater than zero, this means it has a positive cosine value, while the sin of π is less than zero, which means the sine has a negative value.
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One method we use for identifying the sine and cosine values in different quadrants is the CAST diagram that looks like this.
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In the CAST diagram, we know that in the first quadrant, all values are positive.
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In the second quadrant, only the sine value is positive.
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In the third quadrant, only the tangent value is positive.
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And in the fourth quadrant, only cosine is positive.
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If we have a negative sine value and a positive cosine value, we can eliminate quadrant one as all values must be positive there.
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We can eliminate quadrant two as sine is positive there.
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In quadrant three, sine is negative, but so is cosine.
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And that means quadrant three will not work.
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In quadrant four, cosine is positive and sine is negative.
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And that means our angle π under these conditions must fall in the fourth quadrant.
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Letβs consider another example.
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In which quadrant does π lie if sin of π equals one over the square root of two and cos of π equals one over the square root of two?
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When we think about sine and cosine relationships, we know that sin of π is the opposite over the hypotenuse, while the cos of π is the adjacent side over the hypotenuse.
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But how do we translate that information into a coordinate grid?
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In a coordinate grid, the sine, cosine, and tangent relationships will have either positive or negative values.
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And we can remember where each of these relationships will have positive values with the CAST diagram that looks like this.
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In quadrant one, all three trig relationships are positive.
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In quadrant two, only the sine relationship is positive.
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In quadrant three, only the tangent relationship is positive.
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And in quadrant four, only the cosine relationship is positive.
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In this case, weβre dealing with a positive sine relationship and a positive cosine relationship.
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We could also use the information weβre given to find the tangent relationship, which would equal the opposite over the adjacent.
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For this angle, that would be one over one.
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And what weβre seeing is that all three of these relationships are positive for this angle.
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And that means we must say it falls in the first quadrant.
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In our next example, weβll consider an angle thatβs larger than 360 degrees.
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Is cos of 400 degrees positive or negative?
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To answer this question, we need to figure out where 400 degrees would fall on a coordinate grid.
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If we label our standard coordinate grid from zero to 360 degrees, we need to think about what we would do with 400 degrees.
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Traveling counterclockwise one full rotation, weβve gone 360 degrees.
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But in order to get to 400, weβll need to go an additional 40 degrees, since 400 minus 360 equals 40.
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And that means the angle 400 would fall at the same place that the angle 40 degrees falls, here.
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Now weβve 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.
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And to do that, we can use our CAST diagram that looks like this.
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Our CAST diagram tells us where trig relationships are positive in a coordinate grid.
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In the first quadrant, all three relationships are positive.
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In the second quadrant, only sine is positive.
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In the third quadrant, only tangent is positive.
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And in the fourth quadrant, only cosine is positive.
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Our angle falls in the first quadrant.
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In the first quadrant, sine, cosine, and tangent are positive.
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And that means the cos of 400 degrees will be positive.
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Before we finish, letβs review our key points.
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We can identify whether sine, cosine, and tangent will be positive or negative based on the quadrant in which their angle lies.
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In quadrant one, the sine, cosine, and tangent relationships will all be positive.
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For angles falling in quadrant two, the sine relationship will be positive, but the cosine and tangent relationships will be negative.
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For angles falling in quadrant three, the sine and cosine relationships will be negative, but the tangent relationship will be positive.
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And finally, in quadrant four, the sine relationship is negative, the cosine relationship is positive, and the tangent relationship is also negative.
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We often use the CAST diagram to remember this.
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These letters help us identify which values will be positive in which quadrant.