If Sin x= 3 cos x, then find (sinx)(cosx)

[PeacockV30]

I need help with various problems... if anyone could help with ANY of these, I would GREATLY appreciate it!!!

a.) A parabola y=ax^2+bx+c has a vertex (4,2). If (2,0) is on the parabola, then find abc.

b.) Simplify Completely: sin(x-y) cos y + cos(x-y) sin y

c.) The vertices of a parallelogram are (8,5) (-4,11) and (1,-14). The set of points that can serve as a fourth vertex all lie on a circle with center C. Find the exact, simplified coordinates of C

d.)If Sin x= 3 cos x, then find (sinx)(cosx)

e.) If 2137^753 is multiplied out, what is the units digit in the final product?


THANK YOU SO MUCH IF YOU EVEN ATTEMPT ONE OF THESE QUESTIONS! IT HELPS A LOT... IT IS EXTRA CREDIT IN MY CALCULUS CLASS, AND I NEED ALL OF THE HELP THAT I CAN GET!

 

[jsbeckton]

have you tried any of them?

 

[jsbeckton]

a) is pretty much an algebra question assuming that you know the formula for a parabala.
b) is a simple identities question,
c) only one set of points satisfy that; think about triangles and the midpoint and distance formulas.
d) is just elementry trig
e) ? lost me

 

[stapel]

a) One method: You have the vertex (4, 2), with necessarily is midway between the x-intercepts, if any. You have one x-intercept, (2, 0). Find the other x-intercept. Plug these three points into the parabola equation, and solve the resulting system of equations for the values of a, b, and c. Multiply to find the requested value.

b) As suggested, use identities. If you haven't memorized the angle-difference identities, they should be listed in your text.

c) What have you tried?

d) Divide through by cos(x). What do you get?

e) What is the last digit of 2137⁰? What is the last digit of 2137¹? What is the last digit of 2137²? What is the last digit of 2137³? What is the last digit of 2137⁴? Keep going until you see a pattern, and then figure out where "753" fits in that pattern.

Hope that helps a bit. If you get stuck, please reply showing what you have attempted thus far. Thank you.

Eliz.

 

[PeacockV30]

I have tried them, but I struggle with algebra... hence the reason I asked for help with the trig identities...

Eliz, I see some sort of pattern, involving the last digit go up and down by 3, but I do not understand where the 753 comes into play?

 

[stapel]

e) Follow the pattern. It should cycle. Figure out what the cycle is, and then figure out where "753" fits into that. (The "753" comes from the exercise you posted; it's the exponent for which you're asked the last digit in the expansion.)

Eliz.