Math 1330, Chapter 6, Section 1
Lecture A
Popper 15 today
Quiz 11 – split between today and next lecture on double and half-angles
Test 4 material
1
Addition Formulas
Reviewing functions:
Is there any reason to believe that f(5) = f(2) + f(3)? Let’s check with some familiar
functions:
Try it with f (x)  x 2
Try it with f ( x )  x
So it’s not true in general, just for a few special functions.
2
and specifically, it’s not true for trigonometric functions unless it is carefully staged:
sin(
2



)  sin( )  sin( )  2 sin( )




Let’s work out what the RHS of the equation is and compare it to the LHS. Then let’s
look at the graphs on a single set of axes.
3
Popper 15, Question 1
4
There are some formulas – which we’ll give you on Test 4 and the Final –
that you should use when you’ve got added or subtracted inputs:
sin (s + t) = sin s cos t + cos s sin t
sin (s  t) = sin s cos t  cos s sin t
cos (s + t) = cos s cos t  sin s sin t
cos (s  t) = cos s cos t + sin s sin t
tan s  tan t
1  tan s tan t
tan s  tan t
tan ( s  t) =
1  tan s tan t
tan (s + t) =
Note that these formulas work whether your domain value is measured in degrees or in
radian measure.
Now let’s look at applying these formulas
a
12
5
What is sin (

)?

How would you handle
5
5
? What is tan
?


6
How would you handle
7
?
12
Popper 15, Question 2
7
Now let’s get some practice with arbitrary domain values.
If sin x = 2/5 and x is in Quadrant 2 and cos z = 1/3 and sin z < 0,
what is the sin (x + z)?
sin (s + t) = sin s cos t + cos s sin t
8
Popper 15, Question 3
9
Now for more practice:


What is cos( x  )  cos( x  ) ?
3
3
cos (s + t) = cos s cos t  sin s sin t
cos (s  t) = cos s cos t + sin s sin t
10
Popper 15, Question 4
11
Given
sin(75) cos(30)  cos(75)sin(30)
Simplify this formula
sin (s + t) = sin s cos t + cos s sin t
sin (s  t) = sin s cos t  cos s sin t
What is the exact value of cos(
41
)?
12
12
Popper 15, Question 5
13
Given
cos   
cos  
5
7
3
11

2
 
3
   2
2
What is sin(   ) ?
14
Given
tan x  3

What is tan( x  ) ?
3
15
Popper 15, Question 6
Popper 15, Question 7
Simplify the following expression:
sin (s + t) = sin s cos t + cos s sin t
sin (s  t) = sin s cos t  cos s sin t
cos (s + t) = cos s cos t  sin s sin t
cos (s  t) = cos s cos t + sin s sin t
16
Popper 15, Question 8
Simplify the following:
sin (s + t) = sin s cos t + cos s sin t
17
Popper 15, Question 9
Popper 15, Question 10
18