Weekly Tutorial 1 for Math. 69.007A,
Introductory Calculus I

1. Express y = sqrt(3) sin(x) - cos(x) in the form y = a cos(x+b) for suitable values of the numbers a and b. What are a and b?
[Recall sqrt is the square root function...]

a) a = -2, b = Pi/3 [Pi=3.14159...]
b) a = sqrt(3), b = 2 Pi
c) a = 1, b = Pi
d) a = 2, b = Pi/3

2. Solve the equation tan(4x) - tan(2x) = 0 for 0 < x < Pi.

a) x = 0
b) x = Pi/2
c) x = Pi
d) x = Pi/4

3. If sin(x) = 2/3, 0 < x < Pi/2, what is sin(4x) ?

a) 4 sqrt(5)/81
b) 8 sqrt(5)/81
c) sqrt(5)/3
d) 5/9

4. Evaluate sin(2x) given cos(x) + sin(x) = 2/3.

a) -1
b) 1
c) 4/9
d) -5/9

5. Simplify the expression cot(2x) + tan(x) - csc(2x) to its simplest possible form.

a) sin(2x)
b) 0
c) tan(x)
d) cot(x)

Solutions are given within the box: 1. a ...Use the cos-sum formula and compare terms 2. b ...Feed in the values one-by-one and check! 3. b ...Find cos(x), then sin(2x), then cos(2x), then sin(4x) 4. d ...Square both sides, use 2 trig. identities, and solve for sin(2x) 5. b ...Simplify all terms in terms of sin and cos