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Write the expression as the sine cosine, or tangent of an angle. Then find Ihe exact value of the expression sin 75" cos 159 cos 75" sin 150Write the expr...

Question

Write the expression as the sine cosine, or tangent of an angle. Then find Ihe exact value of the expression sin 75" cos 159 cos 75" sin 150Write the expression as the sine cosinetangent of an anglesin 750 cos 150 cos 758 sin 158 (Type an exact answer; using as needed Use integers or fractions for any numbers in the expression )FInd tho exact value of the expression_sin 75 cos 150 cos 759 sin 150 (Simplily your answer: Type an exact answer; using radicals as needed. Use integers or fra

Write the expression as the sine cosine, or tangent of an angle. Then find Ihe exact value of the expression sin 75" cos 159 cos 75" sin 150 Write the expression as the sine cosine tangent of an angle sin 750 cos 150 cos 758 sin 158 (Type an exact answer; using as needed Use integers or fractions for any numbers in the expression ) FInd tho exact value of the expression_ sin 75 cos 150 cos 759 sin 150 (Simplily your answer: Type an exact answer; using radicals as needed. Use integers or fractions for any numbers the expresslon: Ratlonallze all denominators, Enter - your arswer In each of the answer boxes.



Answers

Use the half-angle identities to find the exact value of each trigonometric expression. $$\sin 75^{\circ}$$

Okay, so it's Oprah. Noting that 75 degrees isn't one to this. Makes our co sign for half are unlikely to be positive. So using or half of Angola's ending, we have co sign of, um, 75 times to touch 1 50 over to. That's equal to the square root of one plus coastline of 1 50 degrees over to So what's go sign of 1 50 that's equal to They could have skirted 3/2 over two. So we get, um, combine our numerator into a sickle faction, so that's two minus square root of three over four, so that gives us the square root of two minus square root of three over to.

Okay, so let's start by noting that 75 feet Greece is, um, in quadrant one citizen E makes or half angle eggs. Entity for time. Positive so we can rewrite sign of 75 degrees as 1 50 over two. I know we can use our half an identity that's the square root of one. Minus fine of what 50 over to. So what sign of 1 50? Actually, it's supposed to be co sign. Sorry about that. To co sign of what? 50? That's one minus negative square root of 3/2 over two. So it's good. Mine are numerator into a single fraction, so we get to plus square root of three over four. Did I gives us the square root of two plus? Describe three all over, too.

Okay, so we want to you right co sign. I want to find the exact value of coastline and 75 degrees. Well, let's recall. That's 75 degrees is 120 degrees minus 45 degrees. And now, using our equation, we can rewrite this as co sign of, um, 1 20 Times Co sign of 45 on the minus sign of 1 20 degrees time sign of 45 degrees. Well, it's no that co sign of 1 20 That's going to be negative. One house co sign of 45 degrees. Now it's the square root of two over two on minus sign of 1 20 That's going to be to square roots of 3/2 and sign off 45 degrees. That's square root of 2/2. So now let's remind this into a single fraction here. We're gonna have the negative square root of two minus the square root of six all over to attempt to, which is sport

We have to kind of a little off course. Esquire, 75 degree minus Sinus square, 75 degree. We know an identity cause off duty tha equal toe course Esquire to down minus sign is qualitative. Then we compare this identity. So this question we get you that it will do 75 degree, so it will be cause off toe multiply very 75 degree on solving it will because off intrepidity we know that we can write it cause off 90 plus 60 degree So it will be minus off. Sign 60 degree and it will be minus off Esquire Rudolph three divided by two and this will be able final answer.


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