Video: Using the Pythagorean Identities to Evaluate a Trigonometric Expression

Find (sin πœƒ + cos πœƒ)Β² given that sin πœƒ cos πœƒ = 6/7.

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Video Transcript

Find sin πœƒ plus cos πœƒ squared given that sin πœƒ cos πœƒ equals six over seven.

The first thing we can do with this question is to write out sin πœƒ plus cos πœƒ squared as two sets of brackets. We can treat this like any other binomial expansion and multiply out these brackets to find the following four terms: sin πœƒ times sin πœƒ plus sin πœƒ times cos πœƒ plus cos πœƒ times sin πœƒ and plus cos πœƒ times cos πœƒ.

Now, we know that the order in which sin πœƒ and cos πœƒ are multiplied together does not matter. This means that cos πœƒ sin πœƒ is the same as sin πœƒ cos πœƒ. Upon realizing this, we see that our two middle terms are in fact the same. And we can, therefore, group them together.

Another thing we can do to tidy up our equation, is to see that sin πœƒ times sin πœƒ is sin πœƒ squared. This can equivalently be written as sin squared πœƒ, the same being true for cos. Let’s rewrite these two terms in our expression. Now that we’ve tidied things up, let’s rearrange our terms.

Looking at the first two terms of our expression, you may recognize one of the Pythagorean identities. This identity states that sin squared πœƒ plus cos squared πœƒ is equal to one. The second thing we may notice is that our question gives us the value for sin πœƒ cos πœƒ.

We can, therefore, perform two substitutions on our expression. First, we can replace sin squared πœƒ plus cos squared πœƒ with one. Next, we can replace sin πœƒ cos πœƒ with six over seven.

To make our addition easier, we can rewrite one as seven over seven so that our denominators match. Multiplying six over seven by two, we find 12 over seven. Adding seven over seven to 12 over seven, we find our answer is 19 over seven.

We can therefore, say that sin πœƒ plus cos πœƒ all squared is 19 over seven when sin πœƒ times cos πœƒ is six over seven.

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