# Video: Using Periodic and Cofunction Identities to Evaluate a Trigonometric Function of a Given Angle

π΄π΅πΆ is a right triangle at π΅. Find cot πΌ given that cot π = 4/3.

02:25

### Video Transcript

π΄π΅πΆ is a right triangle at π΅. Find the cot of πΌ given that the cot of π is four-thirds.

Because we know that π΄π΅πΆ is a right triangle, we can also say that π΄π΅π· is a right triangle. And this means we can identify angle π΄π·π΅ as 90 degrees minus π. It also means we can say that πΌ plus 90 degrees minus π is equal to 180 degrees as we know that π΅πΆ forms a straight line. And then if we subtract 90 degrees from both sides of this equation, we see that 90 degrees equals πΌ minus π. And adding π to both sides tells us that πΌ equals 90 degrees plus π, which means the cot of πΌ is equal to the cot of 90 degrees plus π. And now it seems like weβre getting closer because we know cot in terms of π. Based on our cofunction identity, we know that the tan of 90 degrees minus π equals the cot of π, so we want to rearrange the cot of 90 degrees plus π.

We can rewrite it to be the cot of 90 degrees minus negative π. And then, weβll rewrite this in terms of tangent because the cotangent is the reciprocal of tangent. We can say that this is equal to one over tan of 90 degrees minus negative π and that tan of 90 degrees minus negative π simplifies to the cot of negative π. But now we have one over cot of negative π, which will be equal to the tan of negative π. And since tan of negative π is equal to the negative tan of π, itβs an odd function, which means we simplify to the negative tan of π.

Going back to our diagram, if the cot of π is four-thirds, π΄π΅ equals four, π΅π· equals three, the tan of π is the opposite over the adjacent side length, which here is three-fourths. And we need the negative tangent, which will be negative three-fourths. Weβve shown that the cot of πΌ will be equal to the negative tan of π and itβs negative three-fourths.