Use the sum or difference identity to find the exact value of sin255 degrees.

My answer: (-sqrt(2)- sqrt(6)) / (4)

Find the value of tan (alpha-beta), if cos alpha= -3/5, sin beta= 5/13, 90<alpha<180, and 90<beta<180.

My answer: -33/56

Which expression is equivalent to cos(pi-theta)?

Is it -cos theta?

Which expression is not equivalent to cos 2theta?

a. cos^2 theta- sin^2 theta
b. 2cos^2 theta - 1
c. 1 - 2sin^2 theta
d. 2sin theta cos theta

I would pick D.

correct on all counts, good job

For questions on finding "exact" values for certain trig ratios, a quick way to check your result is to simply evaluate your answer with a calculator, then compare it to the calculator result for the initial trig ratio

e.g. for yours I found (-sqrt(2)- sqrt(6)) / (4) to be -.9659258
and sin 255 to be -.9659258 , not bad eh?

For the first question, to find the exact value of sin 255 degrees using the sum or difference identity, we can use the fact that sin (180 + theta) = -sin theta.

Since 255 degrees is greater than 180 degrees, we can rewrite sin 255 degrees as -sin (255 - 180) degrees.
Now, we have sin (255 - 180) degrees = sin 75 degrees.

To find the exact value of sin 75 degrees, we can use the sum or difference identity for sin (alpha + beta). In this case, we can rewrite sin 75 degrees as sin (45 + 30) degrees.

Using the sum identity for sin, we have sin (45 + 30) degrees = sin 45 degrees cos 30 degrees + cos 45 degrees sin 30 degrees.

We know the exact values of sin 45 degrees and cos 30 degrees as 1/sqrt(2) and sqrt(3)/2 respectively. We can substitute these values and simplify the expression:

sin 45 degrees cos 30 degrees + cos 45 degrees sin 30 degrees
= (1/sqrt(2)) * (sqrt(3)/2) + (1/sqrt(2)) * (1/2)
= sqrt(3)/(2sqrt(2)) + 1/(2sqrt(2))
= (sqrt(3) + 1) / (2sqrt(2)).

To rationalize the denominator, we multiply both the numerator and denominator by sqrt(2):

[(sqrt(3) + 1) / (2sqrt(2))] * [sqrt(2) / sqrt(2)]
= [(sqrt(3) + 1) * sqrt(2)] / [2 * sqrt(2) * sqrt(2)]
= [(sqrt(3) + 1) * sqrt(2)] / (2 * 2)
= (sqrt(6) + sqrt(2)) / 4.

Therefore, the exact value of sin 255 degrees is (sqrt(6) + sqrt(2)) / 4.

For the second question, to find the value of tan (alpha - beta), we can use the tangent identity tan (alpha - beta) = (tan alpha - tan beta) / (1 + tan alpha tan beta).

Given that cos alpha = -3/5 and sin beta = 5/13, we can find the values of tan alpha and tan beta.

Using the Pythagorean identity, sin^2 alpha + cos^2 alpha = 1, we can solve for sin alpha: sin alpha = sqrt(1 - cos^2 alpha).
Substituting the given value of cos alpha, we have sin alpha = sqrt(1 - (-3/5)^2) = sqrt(1 - 9/25) = sqrt(16/25) = 4/5.

Now we can find tan alpha and tan beta:
tan alpha = sin alpha / cos alpha = (4/5) / (-3/5) = -4/3.
tan beta = sin beta / cos beta = (5/13) / sqrt(1 - (5/13)^2) = (5/13) / (12/13) = 5/12.

Substituting these values into the tangent identity, we have tan (alpha - beta) = (tan alpha - tan beta) / (1 + tan alpha tan beta) = ((-4/3) - (5/12)) / (1 + (-4/3)(5/12)) = (-33/12) / (1 - (20/36)) = (-33/12) / (1 - 5/9) = (-33/12) / (4/9) = -33/12 * 9/4 = -33/4 * 3/4 = -99/16.

Therefore, the value of tan (alpha - beta) is -99/16.

For the third question, you are correct. The expression equivalent to cos(pi - theta) is -cos theta.

For the fourth question, the expression that is not equivalent to cos 2theta is option A, cos^2 theta - sin^2 theta. The correct expression for cos 2theta is option B, 2cos^2 theta - 1.