In which quadrant does 𝜃 lie if sin 𝜃 equals one over root two and cos 𝜃 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. It’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, we’re at the angle two 𝜋.
Now let’s choose any point on the circumference of this circle. Let’s call this ordered pair 𝑎, 𝑏. Creating a right-angled triangle from this point will help us decide if sin 𝜃 and cos 𝜃 for this angle are individually positive or negative.
Now in this right-angled triangle, the hypotenuse must have a value of one since it’s the radius of the circle. Since we chose the 𝑦-coordinate to be 𝑏, the opposite side has a value of 𝑏 units. The adjacent side has a value of 𝑎 units since we chose the 𝑥-coordinate to be 𝑎.
Now we can refer back to our definition of sine and cosine. Sin 𝜃 equals opposite over hypotenuse. In this case, sin 𝜃 equals 𝑏 over one, or just 𝑏. Cos 𝜃 equals adjacent over hypotenuse, which is 𝑎 over one, or just 𝑎. Since both 𝑎 and 𝑏 are positive values, we now know that, for any value of 𝜃 between zero and 𝜋 over two radians, in other words the first quadrant, both sin 𝜃 and cos 𝜃 are positive.
Let’s now look back to our original question. Sin 𝜃 equals one over root two. And cos 𝜃 equals one over root two. Both of our values of sin 𝜃 and cos 𝜃 are positive. Therefore, our angle 𝜃 must fall between zero and 𝜋 over two radians. That’s the first quadrant.