Question Video: Distinguishing between Identity and Equation in Trigonometry Mathematics

Video Transcript

Is tan 𝜃 equals sin of 𝜃 divided by cos of 𝜃 an identity or an equation?

An identity is a relation that is true for all real variables, while an equation is a relation that is true for some of the real values of the variable and not true for others. Sine, cosine, and tangent are part of the trig identities. So let’s take a look at a right triangle.

So here we have a right triangle. And we we’ll call this angle 𝜃. So now let’s label the sides of the triangle based on 𝜃. The side directly across from 𝜃 would be considered the opposite side. The side across from the 90-degree angle, which is the longest side, is called the hypotenuse. And the side right next to 𝜃 would be considered the adjacent side.

So it’s important to have these sides labeled because the sin of 𝜃 represents a relationship between the opposite side and the hypotenuse side. The cos of 𝜃 is the adjacent side divided by the hypotenuse side. And tan of 𝜃 is equal to the opposite side divided by the adjacent side.

So we wanna know is the tan of 𝜃 equal to the sin of 𝜃 divided by the cos of 𝜃 an identity or an equation. So this means this will have to work for any angle 𝜃 except for the right angle, because the right angle will always stay the right angle. It will not change. We will never need to find the measure of that angle.

So let’s begin with replacing sin of 𝜃 with opposite divided by hypotenuse and now replace the cos of 𝜃 with adjacent divided by hypotenuse. So to simplify this, we are dividing fractions. So we will take the fraction on the bottom, flip it, and actually multiply the top fraction by it. So we take our top fraction and multiply by the reciprocal of the bottom fraction. And we have the hypotenuse that cancels. And we’re left with opposite divided by adjacent, which is right here, exactly what we had. Therefore, this would be an identity.