What are the three Pythagorean identities for the trigonometri...

Question

What are the three Pythagorean identities for the trigonometric functions?

Accepted Answer

Welcome back everyone. In this problem, we want to see which of the following Pythagorean equations is not an identity. So let's go through our list until we find the one that's not correct. To help us interpret the Python equations, we're dealing with our right triangle. So let's just do a little sketch of the right triangle on the right side of our screen. All pun intended. And let me label the sides a, b, and the hypotenuse c, and the angle x degrees. And for our right triangle, two things we know is true. We know that by the Pythagorean theorem, a squared plus b squared will be equal to c squared and using the trigonometric ratios, we also know that the sine of x equals the opposite side a to the hypotenuse c, the cosine of x equals the adjacent side b to the hypotenuse c, and the tangent of x equals the opposite side a to the adjacent side b. Now, let's test our first one. A the square of sine x plus the square of cosine x equals 0. Now, based on what we have here, if we really think about a, what it's telling us is that the square of sine x, where we know that the sine of x is a divided by c. So, that would be a divided by c r squared, plus the square of cosine x, which would be b divided by c r squared equals 0. So that's what we're testing. We don't know if it equals 0, but let's see if if if it's really an identity if it's true. Now a divided by c squared equals a squared divided by c squared. Likewise, b divided by c squared is b squared divided by c squared. Now, if we look at it, both of them have the same denominator of c squared. So we can rewrite it as a squared plus b squared all divided by c squared. But, what do we know? Remember, let's look back at our right triangle. By the Pythagorean theorem, we know a squared plus b squared equals c squared. So, we can rewrite that in our problem and know that would be c squared divided by c squared. I remember we're still testing. We're not sure if it equals to 0. We're just testing it. C squared divided by c squared equals 1. Is 1 equal to 0? No. That's incorrect. So, that means a choice a would have been the Pythagorean equation that is not an identity. Therefore, our answer is answer choice a. How can we know if we are right? Well, if we look at the rest of the statements, we can deduce if it's correct or not. For example, for statement b, the square of the secant of x minus 1 equals the square of the tangent of x. That's a correct statement because it's derived from the Pythagorean identity. The square of the secant of x equals 1 plus the square of the tangent of x. It's just rearranged to make the tan the square of the tangent of x the subject of the equation. Likewise, in option c, one plus the square of the cotangent of x is indeed equal to the square of the cosecant of x. That's also an identity. And finally, in d, the square of the secant of x minus the square of the tangent of x would give us 1. And, the square of the cosecant of x minus the square of the cotangent of x would also give us 1. Therefore, we know that this is an identity as well. Thanks for watching everyone. I hope this video helped.

What are the three Pythagorean identities for the trigonometric functions?

Key Concepts

Pythagorean Identities

Sine and Cosine Relationship

Tangent and Secant Relationship

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