# Trigonometry Overview Contributed by: Topics:
2. The Unit Circle
3. Trigonometric Functions
4. Larger Angles
5. Graphs of the Trig Functions
6. Trigonometric Identities
7. Solving Trig Equations
1. Preparing for the SAT II
Trigonometry
2. Trigonometry
Trigonometry begins in the right
triangle, but it doesn’t have to be
restricted to triangles. The
trigonometric functions carry the
ideas of triangle trigonometry into a
functions and wave forms.
2
3. Trigonometry Topics
 The Unit Circle
 Trigonometric Functions
 Larger Angles
 Graphs of the Trig Functions
 Trigonometric Identities
 Solving Trig Equations
3
 To talk about trigonometric functions, it is
helpful to move to a different system of
 A radian is the measure of a central angle
whose intercepted arc is equal in length to
4
 There are 2 radians in a full rotation --
once around the circle
 There are 360° in a full rotation
 To convert from degrees to radians or
radians to degrees, use the proportion
o
=
360 2π
5
6. Sample Problems
 Find the degree  Find the radian
measure equivalent measure equivalent

4
o
= o
=
360 2π 360 2π
d 3π 4 210o r
= o
=
360o
2π 360 2π
2πd = 270π 360r = 420π
420π 7π
d = 135o r= = 6
360 6
7. The Unit Circle
 Imagine a circle on the
coordinate plane, with its
center at the origin, and
 Choose a point on the
circle somewhere in
7
8. The Unit Circle
 Connect the origin to the
point, and from that point
drop a perpendicular to
the x-axis.
 This creates a right
triangle with hypotenuse
of 1.
8
9. The Unit Circle
 is the
 The length of its legs are angle of
rotation
the x- and y-coordinates of
the chosen point.
 Applying the definitions of 1
y
the trigonometric ratios to x
this triangle gives
bg
cos θ = =x
x
1
y
sin(θ ) = = y
1
9
10. The Unit Circle
 The coordinates of the chosen point are the
cosine and sine of the angle .
 This provides a way to define functions sin()
and cos() for all real numbers .
y bg x
cos θ = =x
sin(θ ) = = y
1 1
 The other trigonometric functions can be
defined from these.
10
11. Trigonometric Functions
 is the
sin(θ ) = y bg
csc θ =
1
y
angle of
rotation
bg
cos θ =x bg
sec θ =
1
x
1
y
x
bg
tan θ =
y
x
bg
cot θ =
x
y
11
12. Around the Circle
 As that point
moves around the
unit circle into
and IV, the new
definitions of the
trigonometric
functions still hold.
12
13. Reference Angles
 The angles whose terminal sides fall in
quadrants II, III, and IV will have values of
sine, cosine and other trig functions which
are identical (except for sign) to the values
 The acute angle which produces the same
values is called the reference angle.
13
14. Reference Angles
 The reference angle is the angle between
the terminal side and the nearest arm of the
x-axis.
 The reference angle is the angle, with vertex
at the origin, in the right triangle created by
dropping a perpendicular from the point on
the unit circle to the x-axis.
14
Original angle  For an angle, , in
reference angle is

Reference angle  sin() is positive
 cos() is negative
 tan() is negative
15
Original angle  For an angle, , in
reference angle is
-
Reference angle  sin() is negative
 cos() is negative
 tan() is positive
16
 For an angle, , in
reference angle is
2
 sin() is negative
 cos() is positive
Original angle  tan() is negative
17
18. All Star Trig Class
 Use the phrase “All Star Trig Class” to
remember the signs of the trig functions in
Star All
Sine is positive All functions
are positive
Trig Class
Tan is positive Cos is positive
18
19. Graphs of the Trig Functions
 Sine
 The most fundamental sine wave, y=sin(x),
has the graph shown.
 It fluctuates from 0 to a high of 1, down to –1,
and back to 0, in a space of 2.
19
20. Graphs of the Trig Functions
 The graph of cb gh
y = a sinb x −h +is
k determined
by four numbers, a, b, h, and k.
 The amplitude, a, tells the height of each peak and
the depth of each trough.
 The frequency, b, tells the number of full wave
patterns that are completed in a space of 2.
 The period of the function is 2π
b
 The two remaining numbers, h and k, tell the
translation of the wave from the origin.
20
21. Sample Problem

  Which of the following

 equations best describes
 the graph shown?
(A) y = 3sin(2x) - 1
   


  (B) y = 2sin(4x)


 (C) y = 2sin(2x) - 1
  (D) y = 4sin(2x) - 1
 (E) y = 3sin(4x)
21
22. Sample Problem

  Find the baseline between the

high and low points.

 Graph is translated -1
    vertically.


 Find height of each peak.
  Amplitude is 3


 Count number of waves in 2
 Frequency is 2
y = 3sin(2x) - 1
22
23. Graphs of the Trig Functions
 Cosine
 The graph of y=cos(x) resembles the graph of
y=sin(x) but is shifted, or translated, π units to
2
the left.
 It fluctuates from 1
to 0, down to –1,
back to 0 and up to
1, in a space of 2.
23
24. Graphs of the Trig Functions
 The values of a, b, h, and k change the shape
and location of the wave as for the sine.
cb gh
y = a cosb x −h + k
Amplitude a Height of each peak
Frequency b Number of full wave patterns
Period 2/b Space required to complete wave
Translation h, k Horizontal and vertical shift
24
25. Sample Problem
 Which of the following

equations best describes 
the graph? 
 (A) y = 3cos(5x) + 4 
 (B) y = 3cos(4x) + 5
   
 (C) y = 4cos(3x) + 5
 (D) y = 5cos(3x) +4
 (E) y = 5sin(4x) +3
25
26. Sample Problem
 Find the baseline

 Vertical translation + 4

 Find the height of peak 
 Amplitude = 5 
 Number of waves in
   
2
 Frequency =3 y = 5cos(3x) + 4
26
27. Graphs of the Trig Functions
 Tangent
 The tangent function has a
discontinuous graph,
repeating in a period of .
 Cotangent
 Like the tangent, cotangent is
discontinuous.
• Discontinuities of the cotangent
are units left
π of those for
2
tangent. 27
28. Graphs of the Trig Functions
 Secant and Cosecant
 The secant and cosecant functions are the
reciprocals of the cosine and sine functions
respectively.
 Imagine each graph is balancing on the peaks and
troughs of its reciprocal function.
28
29. Trigonometric Identities
 An identity is an equation which is true for
all values of the variable.
 There are many trig identities that are useful
in changing the appearance of an
expression.
 The most important ones should be
committed to memory.
29
30. Trigonometric Identities
 Reciprocal Identities  Quotient Identities
1
sin x = sin x
csc x tan x =
cos x
1
cos x =
sec x cos x
cot x =
sin x
1
tan x =
cot x
30
31. Trigonometric Identities
 Cofunction Identities
 The function of an angle = the cofunction of its
complement.
o
sin x = cos(90 −x )
o
sec x = csc(90 −x )
o
tan x = cot(90 −x )
31
32. Trigonometric Identities
 Pythagorean Identities
 The fundamental Pythagorean identity
sin 2 x + cos2 x = 1
2 2
 Divide the first by sin x 2 1 + cot x = csc x
2 2
 Divide the first by cos x 2 tan x + 1 = sec x
32
33. Solving Trig Equations
 Solve trigonometric equations by following
these steps:
 If there is more than one trig function, use
identities to simplify
 Let a variable represent the remaining function
 Solve the equation for this new variable
 Reinsert the trig function
 Determine the argument which will produce the
desired value
33
34. Solving Trig Equations
 To solving trig equations:
 Use identities to simplify
 Let variable = trig function
Solve for new variable

 Reinsert the trig function
 Determine the argument
34
35. Sample Problem
 Solve 3 −3 sinx −2 cos2 x = 0
 Use the Pythagorean
2
identity 3 −3 sin x −2 cos x=0
• (cos2x = 1 - sin2x)
 Distribute
c h
3 −3 sinx −2 1 −sin2 x = 0
3 −3 sinx −2 + 2 sin2 x = 0
 Combine like terms
1 −3 sinx + 2 sin2 x = 0
 Order terms
2 sin2 x −3 sinx + 1 = 0
35
36. Sample Problem
 Solve 3 −3 sinx −2 cos2 x = 0
 Let t = sin x 2 sin 2 x −3 sin x + 1 = 0
2t 2 −3t + 1 = 0
 Factor and solve. (2t −1)(t −1) = 0
2t −1 = 0 t −1 = 0
2t = 1 t =1
1
t=
2
36
37. Sample Problem
 Solve 3 −3 sinx −2 cos2 x = 0
 Replace t = sin x.
π 5π
 t = sin(x) = ½ when x = or
6 6
π
 t = sin(x) = 1 when x=
2