Mathematical Investigations IV
Name:
Mathematical Investigations IV
Trigonometry - Beyond the Right Triangles
A Sum of Tangents
** Optional problems for extra practice.
Tan   


To find the tangent of the sum (or difference) of two angles, we can use the sine and cosine
formulas already derived:
tan     
sin  +   sin cos  cos sin

cos  +   cos cos  sin sin
Divide each term in the fraction by cos cos (or multiply numerator & denominator
1
by
).
cos cos 
sin
sin  +   cos
tan  +   

cos  +   cos
cos
Therefore,


tan    =
cos cos sin

cos cos cos
cos sin sin

cos cos Źcos
tan  tan
1  tan tan
If we replace  by – we can find a formula for tan( – ):
Don't forget that tan(–) = –tan().


tan    =
tan  tan
1  tan tan
In each formula, it is possible to have values of  and  where the denominator goes to zero.
For example, tan    is undefined if tan  tan   1 .


Give an example of values for  and  for which tan  tan   1 .
For what values is the tangent function undefined, in general?
How does your example illustrate this?


How would you describe the relationship between  and  that exists when tan    is
undefined?
When tan     is undefined?
Trig. 9.1
Rev. F08
Mathematical Investigations IV
Name:
Find exact values without a calculator whenever possible. When angles are requested, use your
calculator to approximate.
1.
Find:
L1
(8, 4)
a. tan  =
b. tan  =


c.
sin  =
d.
sin  =
e.
cos  =
f.
cos  =
L2
(4,  6)




g.
tan    
h.
tan    
i.
tan(–) =
j.
tan    
k.
tan (2) =
l.
tan 2 
m.
Find the slope of line L1:
n.
Find the slope of line L2:
o.
Find an angle between the two lines (i.e. formed by the lines), L1 and L2:
p.
How does your answer in part o relate to your answer in part h? Why?
Trig. 9.2


 
Rev. F08
Mathematical Investigations IV
Name:
**2.
Find:
a. tan  =
L1
(3,10 )


=
e.
L2
f.


tan    
c.
(12,4 )
=
g.
b.
tan  =
d.
tan    

+=
h.

–=
Angles of Inclination
In each of the diagrams below,  is the angle of inclination of line AB.
B(-6,5)
B(3,5)

B
(9, 3)
A

(4, 0)

A (0,-4)
A(3,-1)
For each diagram above, find the slope of line AB, tan , and 
slope =
slope =
slope =
tan  =
tan  =
tan  =
=
=
=
State the relationship between the slope of AB and tan .
Trig. 9.3
Rev. F08
Mathematical Investigations IV
Name:
The previous result gives us an alternate way of writing tan     in terms of the slopes m1 and
m2 of the lines L1 and L2:


tan    =
m1  m 2
1  m1  m2
This gives you the tangent of one of the angles between the two lines.
 
If tan     < 0, then the arctangent will give a negative value for    .
If tan    > 0, then the angle is one of the acute angles between the two lines.
The desired angle
between the two lines will be the opposite of this angle or the supplement of the opposite.


What may be stated about the two lines if tan    is
zero?
undefined?
What is the product of the slopes of two lines, if the lines are perpendicular?
State the connections between the above questions.
For each of the following:
a.
Sketch the line and label the angle of inclination, 
b.
Determine the measure of  , the acute angle formed by the x-axis and the given line
c.
Determine the measure of angle of inclination, 
y  2 x  6
1.
10
y
2.
y
10
3
x5
2
y
x
x
-10
-10
d.
-10
-10
10
10
Under what conditions will  =  ? Describe how to determine  otherwise.
Trig. 9.4
Rev. F08
Mathematical Investigations IV
Name:
Without a calculator, find exact values for each of the following. Check with another student at
your table.
5.
tan 105°
**6.
tan 15°
**7.
tan
5
12
8.
tan
11
12
Simplify and evaluate each of the following.
9.
tan 80  tan 20
1  tan 80tan 20
10.
tan 65  tan 25
1 tan 65tan 25
Find the period of each of the following functions.
**11.
y=
tan(4x)  tan(2x)
1  tan(4x)tan(2x)
x 
x 
tan   tan 
2
3 
**12. y =
x  x 
1  tan  tan  
2  3
Trig. 9.5
Rev. F08