Question Video: Identifying the Sign of a Trigonometric Function of a Given Angle Mathematics

Is cos 400° positive or negative?

02:46

Video Transcript

Is cos of 400 hundred degrees positive or negative?

To answer this question, it’s helpful to recall the unit circle. Remember, this circle has a radius of one, and we can add the following measurements of 𝜃 to our graph by moving in an anticlockwise direction.

We start along the 𝑥-axis at zero degrees. Here 𝜃 has a value of 90 degrees. On half a turn, it’s got a value of 180 degrees, 270 degrees, and when we get back to the start we’ve done a full turn, 360 degrees.

Now the question is asking us to establish whether the value of cos of 400 degrees is positive or negative. So we’ll need to continue moving in an anticlockwise direction at intervals of 90 degrees to find the relevant quadrant that 400 degrees lies in.

360 plus 90 gives us 450 degrees. The next value of 𝜃 is 450 degrees. This means then that 400 degrees lies somewhere in this first quadrant. So we’ll need to decide whether for values of 𝜃 in the first quadrant cos is positive or negative.

Remember, cos 𝜃 is equal to adjacent over hypotenuse. So let’s give the ordered pair that corresponds to an angle of 400 degrees a name. We can call it 𝑎, 𝑏, where both 𝑎 and 𝑏 must be positive numbers since it’s in the first quadrant.

We can then construct a right-angled triangle from this ordered pair. The height of this triangle corresponds to the value of the 𝑦-coordinate; it’s 𝑏. And the width of the triangle corresponds to the value of the 𝑥-coordinate; it’s 𝑎. And of course, we’ve already said that the radius of this circle has a length of one unit, so the hypotenuse of our triangle is one unit. The adjacent side in this triangle is the side immediately next to the angle 𝜃. It’s got a length of 𝑎. And the hypotenuse is the side directly opposite the right angle. That’s got a length of one.

So for a value of 𝜃 that lies in the first quadrant, cos of 𝜃 is equal to 𝑎 over one, which is simply 𝑎. Since we said that 𝑎 must be greater than zero — it’s a positive value — that means that cos of 𝜃 must also be greater than zero since it’s equal to 𝑎. And that’s for all values of 𝜃 that lie in the first quadrant. Since 400 degrees lies in the first quadrant, that must mean that cos of 400 degrees is positive.

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