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Question Video: Using the Pythagorean and Periodic Identities to Evaluate the Cosine Function given the Sine Function and Quadrant of an Angle Mathematics • 10th Grade

Find the value of cos (180ยฐ โˆ’ ๐œƒ) given sin ๐œƒ = โˆ’3/5 where 270ยฐ < ๐œƒ < 360ยฐ.


Video Transcript

Find the value of cosine of 180 degrees minus ๐œƒ given the sign of ๐œƒ is negative three-fifths where ๐œƒ is between 270 degrees and 360 degrees.

So essentially, we are looking for cosine, and weโ€™ve been given sine. So we can use some trig identities to help us. Cosine squared ๐œƒ plus sine squared ๐œƒ is equal to one and we know the sine of ๐œƒ is negative three-fifths. So we will be able to find the cosine. So letโ€™s go ahead and square negative three-fifths. Negative three squared is positive nine and five squared is 25.

So now, we need to subtract nine 25ths from both sides of the equation. Now, when we have one minus nine 25ths, we can make the one 25 25ths. So now we subtract the numerator 25 minus nine and we keep the denominator. So cosine squared of ๐œƒ is equal to 16 25ths. And now we square root both sides. The square root of 16 is four and the square root of 25 is five. So cosine of ๐œƒ is four-fifths.

Now, itโ€™s asking about the cosine of 180 degrees minus ๐œƒ. We could rewrite this. We can separate the 180 degrees to be 90 degrees plus 90 degrees. Now using more trig identities, we have that this is equal to the negative sine of 90 degree minus ๐œƒ. And the way that we got this was letting the 90 degrees minus ๐œƒ just represent a general ๐œƒ. So we have the cosine of 90 degrees plus ๐œƒ equals negative sine of ๐œƒ.

Now, negative sine of 90 degrees minus ๐œƒ is equal to negative cosine ๐œƒ. Because we know by more trig identities, the sine of 90 degrees minus ๐œƒ is equal to the cosine of the ๐œƒ. So the only difference is we had a negative sine on the left, which means weโ€™d also need a negative sine on the right. So if these are equal to each other and we found that the cosine of ๐œƒ was four-fifths, then we can take that and replace that for the cosine of ๐œƒ. So negative cosine of ๐œƒ is equal to negative four-fifths.

So we get that our answer is negative four-fifths. Now weโ€™re also given one other piece of information. So we are also told that ๐œƒ is between 270 degrees and 360 degrees. So this means that weโ€™re in quadrant four, between 270 and 360. And we also know that cosine of ๐œƒ represents our ๐‘ฅ-values and sine of ๐œƒ represent our ๐‘ฆ-values. So if we know that the cosine of ๐œƒ was positive four-fifths, we will be looking at the ๐‘ฅ-value in the fourth quadrant, which is positive.

However, what weโ€™re asked to find was this cosine of 180 degrees minus ๐œƒ, which weโ€™ve found to be equal to negative of our cosine of ๐œƒ. And we know that since itโ€™s in the fourth quadrant, cosine of ๐œƒ was positive. But now we have to attach this negative sine out-front. So this means our final answer is negative four-fifths.

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