Problem 88 - Algebra and Trigonometry

Question

Use words (not an equation) to describe one of the Pythagorean identities.

Step-by-Step Solution

Answer

A Pythagorean identity in trigonometry states that, for any angle, the square of the sine of that angle plus the square of the cosine of the same angle is always equal to one. You can visualize this as a point on a unit circle, where the x and y coordinates are the cosine and sine of the angle, respectively. When you square these coordinates and add them together, you will always get one, no matter where the point is on the circle.

Step 1: Understanding Pythagorean Identities

Firstly, it's important to know what the Pythagorean identities are. In trigonometry, there are typically three Pythagorean identities that are based on Pythagoras' theorem. These identities show relationships involving the square of the sine and cosine of a particular angle.

Step 2: Select and Describe an Identity in Words

We'll describe the first identity, which states that the square of the sine of an angle added to the square of the cosine of the same angle is always equal to one. This is true for any angle.

Step 3: Real Life Example for Better Understanding

To better understand this identity, think of a circle with a radius of one unit. If you pick a point on the circle and draw a line to that point from the center of the circle (forming an angle with the x-axis), the x-coordinate of this point corresponds to the cosine of the angle, while the y-coordinate corresponds to the sine of the angle. Now, if you square the x-coordinate and add it to the square of the y-coordinate, you will always get one, regardless of the location of the point on the circle. That's a geometric interpretation of this identity.