REVIEW FOR THE FINAL
1. Find the exact value of
a) all trig. functions of θ, given sec θ = - 5/4 and tan θ <0
b) cos(α+β), sin 2α, tan (α−β ), cos (α/2), given cos α = 4/5, 0<α<π/2, and cosβ = 5/13, -π/2 <β<0
c) sin[cos-1 (-2/3)-sin-1 (4/5)]
d) tan 105o
e) cos 135o , tan 210o , sin (-240o )
f) sin-1 (-1/2), cos-1 ( 2 / 2) , tan-1 (- 3)
2. Graph the function
a) y = -2cos(2x - π/2). What is its amplitude, period and phase shift?
b) y = 3tan(2x + π/4). What is its period?
3. Establish following identities
1 − 2 sin 2 θ
= cot θ − tan θ
a) sin θ cos θ
b)
cos θ
1
=
cos θ − sin θ 1 − tan θ
cosθ − cos(5θ )
= tan( 2θ )
sin θ + sin( 5θ )
4. Solve the following equations on [0,2π)
a) sin2θ = -1
b) cos 2θ = sinθ c) cos θ + sin θ = 2
5. Solve the triangle with
a) a = 6, b = 8, α = 35o
b) a= 4, b= 3, c = 6
6. A radio tower 500 ft high is located on the side of a hill with an inclination to the horizontal of 5o . How
long should two guy wires be if they are to connect to the top of the tower and be secured at two points 100ft
directly above and below the base of the tower.
c)
7. Graph and identify the following equations
a) r = 2- 3 sinθ
b) r = 3 cos 3θ
8. If v = -2i + j and w = 4i -3j, find a) 4v -3w, b) ||v + w||, c) the unit vector in the direction of w.
9. Identify each equation and graph it
a) 4x2 -y2 = 8, b) 4y2 +3x -16y +19 = 0, c) 9x2 + 4x2 -18x +8y = 23
d)x2 + y2 -4x + 6y = 3
10. Write the equation of the hyperbola with vertices at (-3,3), (5,3) and focus at (7,3).
11. Write the equation of the parabola with vertex at (2, -3) and focus at (2,-4).
12. What is the length of the arc subtended by the angle 240o in a circle with radius 5? What is the area of the
sector formed by the angle of 240o in a circle with radius 5?
13. Given z = -4 + 2i and w = 3-5i. Compute z w , z / w , 3z - 5w and write them in the standard form.
14. Write the number z = -4 - 4i in the polar form. Use it to compute z4 .
15. Given vectors u and v. Construct vector ½ u -3v
16. Find rectangular coordinates of the point (-3, 5π/3) given in polar coordinates.