1. Trigonometry for Any Angle
Pre Calculus
Trigonometry for Any Angle

2. Right Triangle Trigonometry

3. Fill out some stuff we already know
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
Abbreviation
Reciprocal Function
Co-function
Right Triangle Definition
Unit Circle Definition
Any Angle Definition
Positive Quadrants
Negative Quadrants
Odd or Even
Domain
Range
Period
Inverse
Inverse Domain

4. Fill out some stuff we already know
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
Abbreviation
sin
cos
tan
csc
sec
cot
Reciprocal Function
Co-function
Right Triangle Definition
Unit Circle Definition
Any Angle Definition
Positive Quadrants
Negative Quadrants
Odd or Even

5. Fill out some stuff we already know
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
Abbreviation
sin
cos
tan
csc
sec
cot
Reciprocal Function
Co-function
Right Triangle Definition
Opp/hyp
Adj/hyp
Opp/adj
Hyp/opp
Hyp/adj
Adj/opp
Unit Circle Definition y x
y/x
1/ y
1/x
x/y
Any Angle Definition
Positive Quadrants
Negative Quadrants
Odd or Even

6. Fill out some stuff we already know
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
Abbreviation
sin
cos
tan
csc
sec
cot
Reciprocal Function
Co-function
Right Triangle Definition
Opp/hyp
Adj/hyp
Opp/adj
Hyp/opp
Hyp/adj
Adj/opp
Unit Circle Definition y x
y/x
1/ y
1/x
x/y
Any Angle Definition
Positive Quadrants
1 and 2
1 and 4
1 and 3
Negative Quadrants
3 and 4
2 and 3
2 and 4
Odd or Even

7. Even and Odd Trig Functions
An even function:
f(x) = f(-x)
cos(30o) = cos(-30o)?
cos(135o) = cos(-135o)?
The cosine and its reciprocal are even functions.

8. Even and Odd Trig Functions
An odd function:
f(-x) = -f(x)
sin(-30o) = -sin(30o)?
sin(-135o) = -sin(135o)?
The sine and its reciprocal are odd functions.

9. Even and Odd Trig Functions
An odd function:
f(-x) = -f(x)
tan(-30o) = -tan(30o)?
tan(-135o) = -tan(135o)?
The tangent and its reciprocal are odd functions.

10. Even and Odd Trig Functions
Cosine and secant functions are even
cos (-t) = cos t sec (-t) = sec t
Sine, cosecant, tangent and cotangent are odd
sin (-t) = - sin t csc (-t) = - csc t
tan (-t) = - tan t cot (-t) = - cot t
Add these to your worksheet

11. Unit Circle Review How can we memorize it? Symmetry
For Radians the denominators help!
Knowing the quadrant gives the correct + / - sign

12. Practice… Get out your Unit Circle, Pencil and Paper!
ON YOUR OWN try these…
Write the question and the answer

23. Check your work

34. How did you do??

35. Reference Angles
Let θ be an angle in standard position. Its reference angle is the acute angle θ’ (called “theta prime”) formed by the terminal side of θ and the horizontal axis.

36. Look at these

37. Reference Angles
Let θ be an angle in standard position and its reference angle has the same absolute value for the functions, the sign ( +/ - ) must be determined by the quadrant of the angle.
Quadrant II θ’ = π – θ (radians)
= 180o – θ (degrees)
Quadrant III θ’ = θ – π (radians)
= θ – 180o (degrees)
Quadrant IV θ’ = 2π – θ (radians)
= 360o – θ (degrees)

38. Trigonometry for any angle
Given a point on the terminal side
Let  be an angle in standard position with (x, y) a point on the terminal side of  and r be the length of the segment from the origin to the point r θ
(x,y)
Then….

39. Trig for any angle Add these definitions to summary worksheet
The six trigonometric functions can be defined as
Add these definitions to summary worksheet

40. Your Chart should look like…
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
Abbreviation
sin
cos
tan
csc
sec
cot
Reciprocal Function
Co-function
Right Triangle Definition
Opp/hyp
Adj/hyp
Opp/adj
Hyp/opp
Hyp/adj
Adj/opp
Unit Circle Definition y x
y/x
1/ y
1/x
x/y
Any Angle Definition
y/r
x/r
r/y
r/x
Positive Quadrants
1 and 2
1 and 4
1 and 3
Negative Quadrants
3 and 4
2 and 3
2 and 4
Odd or Even
Odd
Even

41. Evaluating Trig Functions:
Find sin, cos and tan given (-3, 4) is a point on the terminal side of an angle.
Find r
Find the ratio of the sides of the reference angle
Make sure you have the correct sign based upon quadrant θ r
(-3, 4)
Find r. (-3)2 + (4)2= r2
r =5
sin θ = 4/5
cos θ = -3/5
tan θ = -4/3

42. Name the quadrant… A little different twist…
The cosine and sine of the angle are positive 1 The cosine and sine are negative 3 The cosine is positive and the sine is negative. 4 The sine is positive and the tangent is negative 2 The tangent is positive and the cosine is negative. The secant is positive and the sine is negative.

43. Look at this one…
Given tan  = -5/4 and the cos  > 0, find the sin  and sec .
Which quadrant is it in? θ r
(4, -5)
The tangent is negative, and the cosine is positive
Quadrant IV at point (4, -5)
Find r and use the triangle to find the sine and secant

44. And this one…
Let  be an angle in quadrant II such that sin  = 1/3 find the cos  and the tan . θ 3
(x, 1)
Set up a triangle based upon the information given .
Calculate the other side
Find the other trigonometric functions