aisha said:
Given \csc\theta = \frac {-17} {15} where \frac {3\pi} {2} < \theta <= 2\pi and \cot \beta = \frac {-3} {4} where \frac {\pi} {2} <= \beta <= \pi Find the exact value of \cos (\theta-\beta)
I think the theta angle is in quadrant 4, and the beta angle is in quadrant 2. I drew a circle with a cartesian plane and made the two right angle triangles beta and theta. The beta angle's opposite=4, adjacent=-3 and hypotenuse=5 (using pythagorean theorem) The theta angle's opposite= 15, adjacent = -17 and hypotenuse = square root of 514.
I'm not sure If I have done this correctly so far but this is all that I did and now I am stuck ... Please help!
Your problem: Find the cosine of the difference between theta and beta. In other words, if you start at the x-axis, draw a vector at an angle beta, then draw a vector at an angle of theta. What is the angle between the two vectors? (actually, in this case, you only need to find the cosine of that angle)
Note: I only mention the vectors because drawing them in sometimes helps to visualize just what it is that you're doing. (Also, depending how much more math you take, that cosine difference law is eventually going to turn into a dot product, a very important tool for vectors).
You use the cosine difference identity:
cos (\theta - \beta) = cos \theta cos \beta + sin \theta sin \beta
To solve it, you need the cosine and sine of theta, and the cosine and sine of beta. One step at a time:
Theta
You know the csc of theta. You also know theta is in the fourth quadrant where sine is negative and cosine is positive.
It's csc \theta = \frac{-17} {15}.
The cosecant is just the reciprical of the sine. So the sine of theta is:
sin \theta = - \frac{15}{17}
You then used the pythagorean theorem:
sin^2 \theta + cos^2 \theta = 1 and found that the cosine of theta was 8/17. That gives you two of the values you need to plug into the cosine difference equation.
beta
You were given the cotangent of beta and the fact that it was in the second quadrant. From that you found the adjacent side (-3), the opposite side (4), and the hypotenuse (5).
The cosine is the adjacent over hypotenuse; the sine is the opposite over hypotenuse.
cos \beta = - \frac{3}{5}
sin \beta = \frac{4}{5}
That gives you the other two values you need to plug into your equation.
The cosine of the difference
You have a fraction for the cosine of theta and a fraction for the cosine of beta. Multiply the two fractions.
You also have a fraction for the sine of theta and a fraction for the sine of beta. Multiply those two fractions.
Add the two products together. Both products are guaranteed to have the same denominator, so the addition part is pretty easy. Both numbers you're adding are negatives, so your final answer is negative, as well.
It's just a table of signs,for Christ's sake.