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Problem 13 (6 marks): For an angle 0, we have sin(0) = 0.4.Knowing that cos(0) < 0,find cos(0) and tan(0) without calculating the angle 0....

Question

Problem 13 (6 marks): For an angle 0, we have sin(0) = 0.4.Knowing that cos(0) < 0,find cos(0) and tan(0) without calculating the angle 0.

Problem 13 (6 marks): For an angle 0, we have sin(0) = 0.4.Knowing that cos(0) < 0,find cos(0) and tan(0) without calculating the angle 0.



Answers

In Problems $13-20$, find the exact value without a calculator using half-angle identities. $$ \cos \frac{5 \pi}{12} $$

You want to evaluate the tangent of 75° using half angle identities. So this is the same thing as well. At 70 five degrees, tangent will be positive, so it will be the positive square root of one minus the co sign of. Yeah, let me break this first. The tangent of 150 over to which is equal to the positive square root of one minus the co sign of 1 50/1 plus the co sign of 1 50 which is equal to the sign of 1 50 over one Plus co sign of 1 50. And so at 150° The sign of 1 50 is one half and the co sign of 150 is negative square root 3/2. That would be 1/2 over to get a common denominators three to over two. So would be to minus Square root 3/2 and that would be equal to one half times. To over two minus square root three. And so we get 1/2 minus square root three multiplied by the conjugated to plus square root three, and we get two plus square root three over four -3, which is one, so it's just two plus square root three.

We want to exactly evaluate the CO. Sign a pile or 12 using having no identity. So this is the same thing as pi over six over to the coastline of that. Yeah. And pi over 12 is in the first quadrant. So that's when co sign is positive. So it would be equivalent to the positive square root of one plus the cosine of pi over six over to. And the co sign of pi over six is the spirit of 3/2. This is the same thing as the square root of one half plus the square root of 3/4. Or if we get a common denominator, the spirit of two plus disparity 3/4 which is equal to the square root of two plus the spirit of 3/2.

We want to evaluate the sign of negative seven. High over eight with having their identities. So a negative seven pi over eight. We are in the third quadrant in sign is negative. So keep this in mind when we think of this as the sign of negative seven pi over 4/2 in the half angle identity. First sign taking the negative square root of one minus the co sign of negative seven pyre or four over to negative seven pi over four um is going to be for co sign the square to 2/2. This is equivalent to the negative square roof of one half minus. Discredit to over four, which then we can think of one half as to force and we can think of this as the negative square root of two minus a spirited to over four which would be equivalent to the negative square root of two minus the square root of 2/2.

You want to evaluate the tangent of 15° using half angle identities. This is equivalent to the tangent of 30° over to which is the same thing as 1- the co sign. 30 over The sign of 30. The co sign of 30° is Spirit 3/2. In the sign of 30 is one half. We can simplify the numerator, we can think of one as to over two so that it's two minus square root 3/2 by two by one half, or multiplied other Super bowl to over one. And so we get to minus square root three. Yeah.


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