1. A Review of Trigonometric Functions

2. Right Triangle Vocabulary
hypotenuse c
opposite
adjacent a A C
adjacent
opposite b

3. Trigonometric Functions
Defined in terms of right triangles
sin(x) = opp/hyp
cos(x) = adj/hyp
tan(x) = opp/adj = sin(x) / cos(x)
Know the graphs

4. Trigonometric Functions
Defined in terms of the unit circle
P(x)=(cos x, sin x) 1
sin x x
cos x 1

5. Other Trig Functions cot(x) = 1/tan(x) = cos(x) / sin(x)
sec(x) = 1/cos(x)
csc(x) = 1/sin(x)

6. Odd/Even
Odd
Sin(x)
Csc(x)
Tan(x)
Cot(x)
Even
Cos(x)
Sec(x)

7. Radians
Radian measure of the angle at the center of a unit circle equals the length of the arc that the angle cuts from the unit circle.  1 C

8. Radians s r =  1 s s r =   1 r C Note: Radian measure
is a dimensionless
number

9. Radians and Degrees s 2 r =  2 radian measure 2 arclength
circumference
degree measure
360° = =
2 = 360°
 = 180°

10. Famous Values Angle 0º = 0 30º = /6 45º = /4 60º = /3 90º = /2 Sin1 2 3 4 /2 =0
=1/2 =1
Cos 4 3 2 1 /2 =1
=1/2 =0
Tan
1/ 3 1 3
Und

11. Domain, Range, Period sin(x) cos(x) tan(x) (-, ) x /2, 3/2, ...
(-1, 1)
(-, )
Period 2 

12. Finding the Period Sin (3πx/2 + 4)
Set term that includes x equal to the period of the trig function
3πx/2 = 2π
Solve for x
x = 4/3 = Period

13. Trig Identities to Know
Pythagorean Identities
sin2 x + cos2 x = 1
tan2 x + 1 = sec2 x
cot2 x + 1 = csc2 x
Double Angle
sin(2x)=2sin(x)cos(x)
cos(2x)=cos2(x) – sin2(x)
Square
sin2(x) = (1 – cos(2x))/2
cos2(x) = (1 + cos(2x))/2

14. Creating Inverse Trig Functions
The trig functions are not 1-1
Restrict their domains
y = sin(x) -π/2 ≤ x ≤ π/2
y = cos(x) 0 ≤ x ≤ π
y = tan(x) -π/2 < x < π/2

15. The Inverse Trig Functions
y = sin-1 x or y = arcsin(x)
Domain: [-1, 1]
Range: [-π/2, π/2]
y = cos-1 x or y = arccos(x)
Range: [0, π]
y = tan-1 x or y = arctan(x)
Domain: (-∞, ∞)
Range: (-π/2, π/2)

16. Examples sin x = 0.455 x = sin-1 (0.455) cos x = π/2 x = cos-1 (π/2)
tan x = 8
x = tan-1 (8)