Download Trig Function Accuracy - Madison Area Technical College

Tech Math 2
Angle Precision
1
0.1 or 10
0.01 or 1
Test #01 Review
Angles and Accuracy of Trig Functions
Trig Function Accuracy
Example
2 sig. figs.
sin17  0.29 , tan85  11
3 sig. figs.
sin1.1  0.0192 ,
cos520'  0.996
4 sig. figs.
cos 45.00  0.7071 ,
tan8013'  5.799
Calculating Other Trig Functions
Definition
Example
cos
1
1
cot  

cot 82.3 
 0.135
sin  tan 
tan 82.3
1
1
sec  
sec 22 
 1.1
cos 
cos 22
1
1
csc  
csc 72.67 
 1.048
sin 
sin 72.67
Section 4-4: The Right Triangle
SOH CAH TOA:
opp
hyp
opp
tan  
adj
1
hyp
sec  

cos 
adj
sin  
Page 1 of 4
Pythagorean Theorem: a2 + b2 = c2
adj
cos 
hyp
1
adj
cot  

tan  opp
1
hyp
csc 

sin  opp
Tech Math 2
Test #01 Review
Page 2 of 4
In solving a triangle, you are given three parts of a triangle (one of which must be the length of a side), and
then are expected to calculate the other three parts.
The sum of the angles of a triangle is 180, so for a right triangle (one angle 90), the two acute angles add up to
the remaining 90 (mathematical lingo: they are complementary).
To solve a right triangle:
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 4.5: Applications of Right Triangles
To solve applied right triangle problems:
1. Make a sketch of the situation.
2. Identify/draw right triangles on your sketch that connect given information to unknown information.
3. Solve the right triangle or triangles.
Section 8.4: Applications of Radian Measure
1. A radian is a unit of angle measurement. There are 2 radians in a full circle, as opposed to 360. This
gives us a conversion factor:
 2 radians 
a. To convert from degrees to radians: 57.0 
  0.995 radians
 360 
 360 
b. To convert from radians to degrees: 0.200 radians 
  11.5
 2 radians 
2. Arc Length Formula: s =  r.
a. Note: Lines of latitude are labeled by the angle that is formed from the latitude to the center of
the earth to the equator.
1
3. Area of a Sector of a Circle: A   r 2 .
2

4. Angular Velocity:  
t
Tech Math 2
Test #01 Review
Page 3 of 4
5. 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
Section 9.2: Components of Vectors
1. Components of a vector are two vectors that, when added together, have a resultant equal to the original
vector. (Usually, the components are perpendicular to each other and along the x and y axes, and are
thus called the x- and y- components of a vector.)
2. Resolving a vector into components is calculating the components of a vector.
3. Steps for resolving a vector into x- and y- components:
a. Place the vector A such that its angle is in standard position (i.e., measured counterclockwise
from the x- axis).
b. Calculate the x- and y- components using right triangle trigonometry (i.e., Ax = Acos , and Ay =
Asin , note: these formulas only work if the angle is in standard position!).
c. Check the components for correct sign and magnitude.
Section 9.3: Vector Addition by Components
1. To add vectors by components:
a. Resolve all vectors into components.
b. Add all x-components to get the x-component of the resultant vector (Rx).
c. Add all y-components to get the y-component of the resultant vector (Ry).
d. Find the magnitude of the resultant vector R using the formula R  Rx2  Ry2 .
 Ry 
 , and then
e. Find the angle of the resultant vector R by first using the formula   tan 1 
 Rx 


using the signs of Rx and Ry to convert the angle into the correct quadrant.
Section 9.4: Applications of Vectors
1.
2.
3.
4.
In this section, translate the word problem into vectors, then do vector addition.
Physics: sum of forces = 0 for a body at rest or moving at constant velocity.
Physics: sum of forces = mass times acceleration for accelerating bodies.
To find the displacement between two vectors, subtract vector components instead of adding them!
Tech Math 2
Test #01 Review
Page 4 of 4
Section 9.5: Oblique Triangles, 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
For the ambiguous case, the side opposite the given angle must be less that the side adjacent to the given angle:
B
sin B sin 30

3.875
3.125
Angle B for this triangle is 180 - B from previous
triangle.
Section 9.6: The Law of Cosines
The Law of Cosines
a 2  b 2  c 2  2bc cos A  cos A 
b2  c 2  a 2
2bc
OR
a 2  c2  b2
b  c  a  2ca cos B  cos B 
2ac
2
2
2
OR
c 2  a 2  b 2  2ab cos C  cos C 
a 2  b2  c2
2ab