Video: Finding an Angle Using Values of Trigonometric Functions

In which quadrant does πœƒ lie if sin πœƒ = 1/√2 and cos πœƒ = 1/√2?

02:43

Video Transcript

In which quadrant does πœƒ lie if sin πœƒ equals one over root two and cos πœƒ 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œ‹ over two radians. The second point corresponds to the angle πœ‹. The third corresponds to the angle three πœ‹ over two. And when we get back to the start, we’re at the angle two πœ‹.

Now let’s choose any point on the circumference of this circle. Let’s call this ordered pair π‘Ž, 𝑏. Creating a right-angled triangle from this point will help us decide if sin πœƒ and cos πœƒ 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 π‘Ž units since we chose the π‘₯-coordinate to be π‘Ž.

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

Let’s now look back to our original question. Sin πœƒ equals one over root two. And cos πœƒ equals one over root two. Both of our values of sin πœƒ and cos πœƒ are positive. Therefore, our angle πœƒ must fall between zero and πœ‹ over two radians. That’s the first quadrant.

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