Video: Using the Pythagorean Identities to Evaluate a Trigonometric Function for an Angle

Find the value of secΒ² πœƒ given tanΒ² πœƒ = 4.

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Video Transcript

Find the value of sec squared πœƒ given tan squared πœƒ equals four.

To answer this question, we’re not going to solve the equation tan squared πœƒ equals four for πœƒ. Instead, we’re going to find a Pythagorean identity that links sec squared πœƒ and tan squared πœƒ.

To find this identity, we’re going to recall one of the most straightforward trigonometric identities. That is, sin squared πœƒ plus cos squared πœƒ equals one. We also know that tan πœƒ is equal to sin πœƒ over cos πœƒ. So it follows that tan squared πœƒ must also be equal to sin squared πœƒ over cos squared πœƒ.

And so what we’re going to do is individually divide each of the elements in our first identity by cos squared πœƒ. When we do, we can write sin squared πœƒ divided by cos squared πœƒ is tan squared πœƒ. Cos squared πœƒ divided by cos squared πœƒ is one. And since one over cos πœƒ is sec πœƒ, we know one over cos squared πœƒ is sec squared πœƒ.

And so this is the Pythagorean identity we’re going to use. And in fact, we don’t need to derive it each time. We can simply quote that sec squared πœƒ is equal to tan squared πœƒ plus one. In this question, tan squared πœƒ is equal to four. So we can say that sec squared πœƒ must be equal to four plus one, which is simply five. And so the value of sec squared πœƒ given that tan squared πœƒ is equal to four is five.

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