Download Tech Math 2 Lecture Notes - Madison Area Technical College

Math Analysis Test #04 Review Sheet
Friday 10/06/2006
Page 1 of 8
Section 6,1: Angle Measure
 180 
1. (radians)  
 = (degrees)



  
2. (degrees)  
 = (radians)
 180 
3. Arc Length: s = r.
4. Sector Area: A = ½r2.

5. Angular Speed:   .
t
6. Some units of angular velocity:
radians radians radians
a.
,
,
second minute
hour
degrees degrees degrees
b.
,
,
second minute
hour
revolutions revolutions
revolutions
c.
,
(rpm) ,
second
minute
hour
7. Speed of an object moving around a circle: v =  r.
Section 6.2: Trigonometry of Right Triangles
Big Idea: The six trigonometric functions we have studied can be applied to the ratios of sides in any right
triangle to find unknown quantities of those right triangles.
Big Skill: You should be able to identify known and unknown measurements on a right triangle and relate
those quantities using an appropriate trigonometric function.
SOH CAH TOA:
opp
hyp
opp
tan  
adj
1
hyp
sec  

cos 
adj
sin  
To solve a right triangle:
adj
hyp
1
adj
cot  

tan  opp
1
hyp
csc 

sin  opp
2
2
2
a b  c
cos 
Math Analysis Test #04 Review Sheet
Friday 10/06/2006
Page 2 of 8
1. Make a sketch of the triangle, label sides and angles consistently (a, b, and c for the legs and
hypotenuse; A and B for the complementary angles), and label the given information.
2. Find a way to relate the unknown parts to the given information using a trig function (sine, cosine, or
tangent) or the Pythagorean Theorem (a2 + b2 = c2). Try to use original given information to minimize
rounding errors.
3. Check your work:
a. Make sure the sides obey the Pythagorean Theorem.
b. Make sure the angles add up to 180.
c. Make sure unused trig functions give the right answers.
d. Make sure that the longest side is opposite the largest angle, and the shortest side is opposite the
smallest angle.
Section 6.3: Trigonometric Functions of Angles
Definition of the Six Trigonometric Functions: Let  be the angle to the terminal point P(x, y). If
r  x 2  y 2 is the distance from the origin to the terminal point P, then:
y
a. sin  
r
x
b. cos  
r
y
c. tan  
x
x
d. cot  
y
r
e. sec  
x
r
f. csc 
y
Area of a triangle: A 
1
ab sin 
2
Math Analysis Test #04 Review Sheet
Friday 10/06/2006
Page 3 of 8
Section 6.4: The Law of Sines
The Law of Sines
a
b
c


sin A sin B sin C
OR
sin A sin B sin C


a
b
c
The Law of Sines uses the sine function to relate the lengths of the sides of any triangle to its angles. The Law
of Sines is used when:
1. One side and two angles are given (the AAS, SAA, or ASA cases).
2. Two sides and an angle opposite one of the sides is given (the ASS or SSA cases).
This can sometimes lead to the so-called “ambiguous case.” The ambiguous case occurs when a given
side opposite the given angle is greater than the sine of that angle times the given side adjacent to the
angle, but less than the adjacent side. When this is the case, you get the other angle by subtracting the
angle you get from 180.
Section 6.5: The Law of Cosines
The Law of Cosines
a  b 2  c 2  2bc cos A
2
cos A 
OR
OR
b 2  c 2  a 2  2ca cos B
OR
a 2  c 2  b2
cos B 
2ac
c 2  a 2  b 2  2ab cos C
OR
cos C 
Heron’s Formula: A  s  s  a  s  b  s  c  , s 
abc
2
b2  c 2  a 2
2bc
a 2  b2  c 2
2ab
Math Analysis Test #04 Review Sheet
Friday 10/06/2006
Page 4 of 8
Section 7.1: Trigonometric Identities
1. Reciprocal Identities:
1
a. csc t 
sin t
1
b. sec t 
cos t
1
cos t

c. cot t 
tan t sin t
sin t
d. tan t 
cos t
2. Pythagorean Identities:
g. sin 2 t  cos 2 t  1
h. tan 2 t  1  sec 2 t
a. cot 2 t  1  csc 2 t
3. Even-Odd Identities:
a. sin(t )   sin t
b. cos(t )  cos t
c. tan(t )   tan t
4. Cofunction Identities:


a. sin   t   cos t
2 


b. cos   t   sin t
2 


c. tan   t   cot t
2 
5. To simplify a trigonometric expression, use factoring, common denominators, and trigonometric
identities to convert the expression into one that has fewer trig functions and calculations.
6. To prove a trigonometric identity, try to show how the more complicated simplifies to the simpler side
using identities or by writing everything in terms of sines and cosines. If both sides are complicated, try
to simplify both sides.
Section 7.2: Addition and Subtraction Formulas
1. sin  s  t   sin s cos t  cos s sin t
2. sin  s  t   sin s cos t  cos s sin t
3. cos  s  t   cos s cos t  sin s sin t
4. cos  s  t   cos s cos t  sin s sin t
5. tan  s  t  
tan s  tan t
1  tan s tan t
Math Analysis Test #04 Review Sheet
6. tan  s  t  
Friday 10/06/2006
tan s  tan t
1  tan s tan t
7. A sin x  B cos x  k sin  x    , where k  A2  B 2 and cos  
Section 7.3: Double Angle, Half Angle, and Product-Sum Formulas
1. Double-Angle Formulas:
a. sin 2x  2sin x cos x
cos 2 x  cos 2 x  sin 2 x
b.
 1  2sin 2 x
 2 cos 2 x  1
2 tan x
c. tan 2 x 
1  tan 2 x
2. Power-Lowering Formulas:
1  cos 2 x
a. sin 2 x 
2
1

cos
2x
b. cos 2 x 
2
1

cos
2x
c. tan 2 x 
1  cos 2 x
3. Half-Angle Formulas:
u
1  cos u
a. sin  
2
2
u
1  cos u

2
2
u 1  cos u
sin u

c. tan 
2
sin u
1  cos u
4. Product-to-Sum Formulas:
1
a. sin u cos v  sin  u  v   sin  u  v  
2
1
b. cos u sin v  sin  u  v   sin  u  v  
2
1
c. cos u cos v  cos  u  v   cos  u  v  
2
1
d. sin u sin v  cos  u  v   cos  u  v  
2
5. Sum-to-Product Formulas:
x y
x y
cos
a. sin x  sin y  2sin
2
2
x y
x y
sin
b. sin x  sin y  2 cos
2
2
b. cos
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A
B
and sin  
k
k
Math Analysis Test #04 Review Sheet
Friday 10/06/2006
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x y
x y
cos
2
2
x y
x y
sin
d. cos x  cos y  2sin
2
2
c. cos x  cos y  2 cos
Section 7.4: Inverse Trigonometric Functions
1. All inverse trig problems have two answers; the second answer must be found by looking at the unit
circle, because the domains of the trig functions get restricted to make them one-to-one.
2. The inverse sine function: sin-1(x), or arcsin(x)
sin(x)
Domain: -/2  x  /2 (restricted)
Range: -1  y  1
sin-1(x)
Domain: -1  x  1
Range: -/2  y  /2
Math Analysis Test #04 Review Sheet
Friday 10/06/2006
3. The inverse cosine function: cos-1(x), or arccos(x)
cos(x)
Domain: 0  x   (restricted)
Range: -1  y  1
cos-1(x)
Domain: -1  x  1
Range: 0  y  
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Math Analysis Test #04 Review Sheet
Friday 10/06/2006
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4. The inverse tangent function: tan-1(x), or arctan(x)
tan(x)
Domain: -/2  x  /2 (restricted)
Range: -  y  
tan-1(x)
Domain: -  x  
Range: -/2  y  /2
5. Use a right triangle to compute the answer to a trig function composed with an inverse trig functions.
6. To calculate the inverse trig functions of the “black sheep” functions:
a. sec1    cos 1  1 
b. csc1    sin 1  1 
c. cot 1    tan 1  1 
Section 7.5: Trigonometric Equations
1. To solve trig equations, try to combine all trig functions into one trig function, then isolate it and take
the inverse of both sides of the equation.
2. Also, don’t forget that polynomial equations of trigonometric functions can be solved by factoring.