In which quadrant does 𝜃 lie if sin 𝜃 is greater than zero and cos 𝜃 is greater than zero?
Consider the unit circle. Remember, this circle has a radius of one. And we can add the following values of 𝜃 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 𝜃 if both sin 𝜃 and cos 𝜃 are positive.
Remember, sin 𝜃 is equal to opposite over hypotenuse and cos 𝜃 is equal to adjacent over hypotenuse. Let’s start by choosing a value of 𝜃 that lies in the first quadrant. This value of 𝜃 must therefore be between zero and 𝜋 over two.
We can give this ordered pair a name, 𝑎, 𝑏, where both 𝑎 and 𝑏 are positive real numbers. We can then create a right-angled triangle where the side adjacent to 𝜃 is given as 𝑎 and the side opposite 𝜃 is given as 𝑏.
Since this is the unit circle, we also know that the value of the hypotenuse is one. Let’s substitute the relevant values into our equations for sin 𝜃 and cos 𝜃. Sin 𝜃 is opposite over hypotenuse. In this case, that’s 𝑏 over one, which is just 𝑏.
We stated that the value of 𝑏 must be a positive real number. Therefore, the value of sin 𝜃 in this quadrant is greater than zero. It’s positive. Cos 𝜃 is given as adjacent over hypotenuse. The adjacent in our right-angled triangle is 𝑎 and the hypotenuse is one. So cos 𝜃 is 𝑎 over one, or 𝑎.
Since 𝑎 lies in the first quadrant, 𝑎 must be a positive real number. So cos 𝜃 is greater than 𝑎. It’s also positive. If sin 𝜃 and cos 𝜃 are both greater than zero, 𝜃 lies in the first quadrant.