Question Video: Recognizing Trigonometric Functions from their Graphs Mathematics

Video Transcript

Which function does the plot shown in the graph represent? (A) Sine, (B) cosine, or (C) tangent. Assign each region of the plot with the corresponding quadrant of the unit circle.

Let’s consider the graph we’ve been given. First, we observe that this function cannot be the tangent function because it has no vertical asymptotes. The tangent function has vertical asymptotes at both 𝜋 by two and negative 𝜋 by two radians. But on this graph, this is not the case. So, we can rule the tangent function out, and we’re left with sine and cosine. The graphs of the sine and cosine functions are very similar. They have the same basic shape, and they share a lot of key properties. For example, both are periodic with a period of two 𝜋 and both oscillate between a minimum value of negative one and a maximum value of positive one. We can see these values shown on the vertical axis.

To distinguish between the two functions, we need to consider the roots of the function plotted on the graph. We observe that when 𝑥 is equal to zero radians, the value of the function is also equal to zero. And when 𝑥 is equal to 𝜋 radians, the value of the function is again zero. On the unit circle, an angle of 𝜋 radians corresponds to a half-turn in the counterclockwise direction, which brings us to the negative 𝑥-axis. We recall also that points on the unit circle have the coordinates cos 𝜃, sin 𝜃, where 𝜃 is the counterclockwise angle between the positive 𝑥-axis and the radius joining the point to the origin.

The point on the unit circle, which lies on the negative 𝑥-axis, has coordinates negative one, zero. And we’ve identified that this corresponds to an angle of 𝜋 radians. So, this tells us that the cos of 𝜋 is equal to negative one and the sin of 𝜋 is equal to zero. On the graph we’ve been given, the value of the function is zero when the angle is 𝜋. And so, the graph must represent the sine function.

Let’s now consider the second part of the question, in which we are asked to assign each region of the plot with the corresponding quadrant of the unit circle. Region A first of all corresponds to angles between negative 𝜋 and negative 𝜋 by two radians. On the unit circle, negative angles correspond to clockwise turns. An angle of negative 𝜋 radians is a half-turn clockwise from the positive 𝑥-axis, and an angle of negative 𝜋 by two radians is a quarter-turn clockwise from the positive 𝑥-axis. Angles between these two values would therefore fall in the third quadrant.

Next, let’s consider region B. This corresponds to angles between negative 𝜋 by two radians and zero radians. We’ve already said that an angle of negative 𝜋 by two radians would correspond to a quarter-turn clockwise from the positive 𝑥-axis. And an angle of zero radians would correspond to the positive 𝑥-axis itself. Region B therefore corresponds to the fourth quadrant.

For region C, we’re looking at angles between zero and 𝜋 by two radians. An angle of 𝜋 by two radians — that’s positive 𝜋 by two radians — would correspond to a quarter-turn counterclockwise from the positive 𝑥-axis. An angle of zero radians corresponds to the positive 𝑥-axis itself. So, the angles in region C, angles between zero and 𝜋 by two radians, lie in quadrant I.

The angles in region D, which are between 𝜋 by two and 𝜋 radians, must therefore correspond to the second quadrant. An angle of 𝜋 by two radians we’ve said would take us to the positive 𝑦-axis, and an angle of 𝜋 radians, a half-turn counterclockwise, would take us to the negative 𝑥-axis. So, this does indeed correspond with quadrant II.

So, we found that the function shown in the plot is the sine function and that region A corresponds to quadrant III, region B corresponds to quadrant IV, region C corresponds to quadrant I, and region D corresponds to quadrant II, each of the unit circle.