# Video: Determining the Quadrant in Which an Angle Lies given the Signs of Its Trigonometric Functions

In which quadrant does π lie if sin π > 0 and cos π > 0?

02:24

### Video Transcript

In which quadrant does π lie if sin π is greater than zero and cos π is greater than zero?

Consider the unit circle. Remember, this circle has a radius of one. And we can add the following values of π to our graph by moving in an anticlockwise direction. We start at zero. This point is π over two radians or 90 degrees, π radians, three π over two, and back to two π, or 360 degrees. The question is asking us to determine the value of π if both sin π and cos π are positive.

Remember, sin π is equal to opposite over hypotenuse and cos π is equal to adjacent over hypotenuse. Letβs start by choosing a value of π that lies in the first quadrant. This value of π must therefore be between zero and π over two.

We can give this ordered pair a name, π, π, where both π and π are positive real numbers. We can then create a right-angled triangle where the side adjacent to π is given as π and the side opposite π is given as π.

Since this is the unit circle, we also know that the value of the hypotenuse is one. Letβs substitute the relevant values into our equations for sin π and cos π. Sin π is opposite over hypotenuse. In this case, thatβs π over one, which is just π.

We stated that the value of π must be a positive real number. Therefore, the value of sin π in this quadrant is greater than zero. Itβs positive. Cos π is given as adjacent over hypotenuse. The adjacent in our right-angled triangle is π and the hypotenuse is one. So cos π is π over one, or π.

Since π lies in the first quadrant, π must be a positive real number. So cos π is greater than π. Itβs also positive. If sin π and cos π are both greater than zero, π lies in the first quadrant.