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

𝐴𝐡𝐢 is a right triangle at 𝐡. Find cot 𝛼 given that cot πœƒ = 4/3.

02:25

Video Transcript

𝐴𝐡𝐢 is a right triangle at 𝐡. Find the cot of 𝛼 given that the cot of πœƒ is four-thirds.

Because we know that 𝐴𝐡𝐢 is a right triangle, we can also say that 𝐴𝐡𝐷 is a right triangle. And this means we can identify angle 𝐴𝐷𝐡 as 90 degrees minus πœƒ. It also means we can say that 𝛼 plus 90 degrees minus πœƒ is equal to 180 degrees as we know that 𝐡𝐢 forms a straight line. And then if we subtract 90 degrees from both sides of this equation, we see that 90 degrees equals 𝛼 minus πœƒ. And adding πœƒ to both sides tells us that 𝛼 equals 90 degrees plus πœƒ, which means the cot of 𝛼 is equal to the cot of 90 degrees plus πœƒ. And now it seems like we’re getting closer because we know cot in terms of πœƒ. Based on our cofunction identity, we know that the tan of 90 degrees minus πœƒ equals the cot of πœƒ, so we want to rearrange the cot of 90 degrees plus πœƒ.

We can rewrite it to be the cot of 90 degrees minus negative πœƒ. And then, we’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œƒ and that tan of 90 degrees minus negative πœƒ simplifies to the cot of negative πœƒ. But now we have one over cot of negative πœƒ, which will be equal to the tan of negative πœƒ. And since tan of negative πœƒ is equal to the negative tan of πœƒ, it’s an odd function, which means we simplify to the negative tan of πœƒ.

Going back to our diagram, if the cot of πœƒ is four-thirds, 𝐴𝐡 equals four, 𝐡𝐷 equals three, the tan of πœƒ 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. We’ve shown that the cot of 𝛼 will be equal to the negative tan of πœƒ and it’s negative three-fourths.

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