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

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


Video Transcript

In which quadrant does πœƒ lie if sin πœƒ is greater than zero and cos πœƒ 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œƒ to our graph by moving in an anticlockwise direction. We start at zero. This point is πœ‹ over two radians or 90 degrees, πœ‹ radians, three πœ‹ over two, and back to two πœ‹, or 360 degrees. The question is asking us to determine the value of πœƒ if both sin πœƒ and cos πœƒ are positive.

Remember, sin πœƒ is equal to opposite over hypotenuse and cos πœƒ is equal to adjacent over hypotenuse. Let’s start by choosing a value of πœƒ that lies in the first quadrant. This value of πœƒ must therefore be between zero and πœ‹ over two.

We can give this ordered pair a name, π‘Ž, 𝑏, where both π‘Ž and 𝑏 are positive real numbers. We can then create a right-angled triangle where the side adjacent to πœƒ is given as π‘Ž and the side opposite πœƒ 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 πœƒ and cos πœƒ. Sin πœƒ 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 πœƒ in this quadrant is greater than zero. It’s positive. Cos πœƒ is given as adjacent over hypotenuse. The adjacent in our right-angled triangle is π‘Ž and the hypotenuse is one. So cos πœƒ is π‘Ž over one, or π‘Ž.

Since π‘Ž lies in the first quadrant, π‘Ž must be a positive real number. So cos πœƒ is greater than π‘Ž. It’s also positive. If sin πœƒ and cos πœƒ are both greater than zero, πœƒ lies in the first quadrant.

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