The following formulas will be provided. It is your responsibility to locate the formula
sheet before you start your test. If you can’t find it, ask proctors for help. Handy Formulas ( + ) = + ( ) = ( + ) = ( ) = + ( + ) = tan s tan t
1 tan s tan t ( ) = tan s tan t
1 tan s tan t sin(2) = 2 sin cos (2) = cos 2 t sin 2 t sin s
1 cos s 2
2 s
1 cos s
cos 2
2 tan s
sin s 2 1 cos s Example 1a: Let f ( x) x 2 x 12
be a given rational function.
x2 − 2x 3 a) Domain: b) Vertical Asymptote(s): c) Hole: d) Horizontal Asymptote: e) Does the graph intersect the HA? If so, what is the x-coordinate of the intersection? Example 1b: f ( x) x4
x +3x - 4
2 a) Domain: b) Vertical Asymptote(s): c) Hole: d) Horizontal Asymptote: e) Does the graph intersect the HA? If so, what is the x-coordinate of the intersection? 2 Example 2: For the following functions, find the difference quotient
and simplify the difference quotient for x 1. f ( x h) f ( x )
h a) f ( x ) 2 x 2 7 x 4 b) f ( x ) 3
x2 3 Example 3: Given f ( x ) 4 ln( x ) and g ( x ) e 3 x , find the following:
a) ( f g )(3) b) ( g f )(2) Example 4: A motorcycle has wheels with a 12 inch radius. If each wheel’s rate of turn is 4
revolutions per second, how fast is the car moving in units of inches/sec? Example 5: Find the area of the sector if
a) r 4, 6 b) r 8, 60 0 4 Example 6: Simplify the following expressions:
a) sin( )
sin( ) 1 cos( ) 1 cos( ) b) (1 sin )(sec tan ) c) tan(x) cos(x)
1 sin( x) 5 Example 7: Evaluate 5 11 + sin 4 6 a) tan 2 5 + cos 3 6 b) cot Example 8: Suppose that is an acute angle of a right triangle and that tan( ) a) Find sin( ) and cos( ). 3 2
.
5 b) Evaluate 1 cos2 ( ). c) Evaluate 1 sin 2 ( ). 6 Example 9:
a) In the figure below, angle B is a right angle, m(D)=45 and m(ACB)=60.
If AC=10, find the length of AD.
A B C D b) In the figure below, angle B is a right angle, m(D)=30 and m(ACB)=60. If AC=5,
find the length of AD.
A B C D c) In the figure below, angle B is a right angle, m(D)=30 and m(ACB)=45. If AC=8,
find the length of AD.
A B C D
7 Example 10: Given a triangle ABC, with B being a right angle, and AB =7 and AC=12,
find all six trigonometric functions for angle C. Example 11: Evaluate the following if possible: 5 a) tan 3 5 d) sin 3 5 b) cot 6 4 e) csc 3 11 g) sin 6 7 j) cos 4 3 h) cot 4 m) tan 45 0 s) sin 240 p) sin 45 0
0 k) tan 135 0 q) cos 60 t) sin 210 n) cot 225 0
0 2 c) cos 3 7 f) sec 6 5 i) tan 4 l) cos 360 0 r) cos300 o) tan 315 0
0 0 8 Example 12: Let P(x,y) denote the point where the terminal side of an angle θ meets the
unit circle.
a) If P is in Quadrant II and y=3/4, find
sec(θ)
cot(θ) b) If P is in Quadrant IV and x=2/5, find
csc(θ)
cot(θ) 9 1) Given tan( x) 1
, 90 0 x 180 0 ,
5 a) sin(2 x) b) cos(2 x) 2) Evaluate
a) sin ( 11
3 ) cos ( 11 −31
6 ) tan ( 25 sin(− 3 ) cos(− 6 )
b)
20
tan(
)
3 41
4 ) 3) Solve the following equation on the interval [0,2 ) : 4 sin 2 ( x ) 9 sin( x ) 5 0 4) Find the exact value of the following expression: 3 sin 1 tan1 3
a) cos1 2 2 b) 1 2
2 arcsin arccos 2 arctan 1
2 3 c) tan1 cos1 sin 1 2 2 3 1 1 5) Find the exact value of the following expression:
2 a) sec tan1 5 5 b) cos sin 1 4 5 c) sin tan1 4 6) Find the vertical asymptotes for the following functions: a) f ( x) 2 csc x 4 b) f ( x ) 12 sec x
2 c) f ( x ) 10 tan 2 x c) f ( x ) 2 cot 3x 2 2 7) a) Write a sine function with amplitude 10, horizontal shift 5 to the left, vertical shift
2 down, and period 5. b) Write a cosine function with amplitude 6, horizontal shift 3 to the right, vertical shift 4 up, and period .
3 8) Given the following graph, find the formula for it:
The point C(1/8, 2) is on this graph; x= 1/4 and x=-1/4 are asymptotes. 9) Write the equivalent form
a) 1−2 2 ()
6 sin(−) cos(−) b) −2 2 ( )+1
8 sin(−) cos(−) c) 10 sin(−) cos(−) 2 ()−2 () 10) Given sin( x) a) sin( x y ) b) sin( x y ) c) cos( x y ) d) cos( x y ) 1
2
0
0
0
0
, 90 x 180 , and sin( y) , 180 y 270 , find:
4
5 11) Find an equation for the sine function passing through (0,0) given that the first maximum point on the right of the origin is ,5 .
4 12) a) Find all solutions of the equation b) Find all solutions of the equation 4 cos(3x ) 3 5 2 sin(4 x ) 3 2 on the interval on the interval 2 0, 3 . 0, 2 . 13) a) f ( x) 5 sin(6 x)
What is the domain of this function? What is the range of this function? Graph the function over one period. Label the intercepts, minimum/maximum
value(s) with an ordered pair. b) f ( x) 6 cos(2 x)
What is the domain of this function? What is the range of this function? Graph the function over one period. Label the intercepts, minimum/maximum
value(s) with an ordered pair.

 

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