Video: Finding an Angle Using Values of Trigonometric Functions

In which quadrant does π lie if sin π = 1/β2 and cos π = 1/β2?

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Video Transcript

In which quadrant does π lie if sin π equals one over root two and cos π equals one over root two?

Consider the unit circle. Remember, the length of the line joining the origin to any point on the circle is one unit. Itβs a circle with radius of one. The axes intercepts therefore lie at one, zero; zero, one; minus one, zero; and zero, minus one.

Remember, since we are measuring in an anticlockwise direction, this first point corresponds to the angle π over two radians. The second point corresponds to the angle π. The third corresponds to the angle three π over two. And when we get back to the start, weβre at the angle two π.

Now letβs choose any point on the circumference of this circle. Letβs call this ordered pair π, π. Creating a right-angled triangle from this point will help us decide if sin π and cos π for this angle are individually positive or negative.

Now in this right-angled triangle, the hypotenuse must have a value of one since itβs the radius of the circle. Since we chose the π¦-coordinate to be π, the opposite side has a value of π units. The adjacent side has a value of π units since we chose the π₯-coordinate to be π.

Now we can refer back to our definition of sine and cosine. Sin π equals opposite over hypotenuse. In this case, sin π equals π over one, or just π. Cos π equals adjacent over hypotenuse, which is π over one, or just π. Since both π and π are positive values, we now know that, for any value of π between zero and π over two radians, in other words the first quadrant, both sin π and cos π are positive.

Letβs now look back to our original question. Sin π equals one over root two. And cos π equals one over root two. Both of our values of sin π and cos π are positive. Therefore, our angle π must fall between zero and π over two radians. Thatβs the first quadrant.