1. Trigonometric identities Trigonometric formulae

2. Angles
Trigonometric ratios

3. Angles
Reciprocal ratios

4. Right Triangle Trig DefinitionsB c a A C b
sin(A) = sine of A = opposite / hypotenuse = a/c
cos(A) = cosine of A = adjacent / hypotenuse = b/c
tan(A) = tangent of A = opposite / adjacent = a/b
csc(A) = cosecant of A = hypotenuse / opposite = c/a
sec(A) = secant of A = hypotenuse / adjacent = c/b
cot(A) = cotangent of A = adjacent / opposite = b/a

5. Angles
Pythagoras’ theorem
The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides

6. Angles
Special triangles
Right-angled isosceles

7. Angles
Special triangles
Half equilateral

8. Special Right Triangles
30°
45° 2 1
60°
45° 1 1

9. Angles
Special triangles
Half equilateral

10. Trigonometric identities
The fundamental identity
The fundamental trigonometric identity is derived from Pythagoras’ theorem

11. Trigonometric identities
Two more identities
Dividing the fundamental identity by cos2

12. Trigonometric identities
Two more identities
Dividing the fundamental identity by sin2

14. Trigonometric formulas
Sums and differences of angles
Double angles
Sums and differences of ratios
Products of ratios

15. Trigonometric formulas
Sums and differences of angles

16. Trigonometric formulas
Double angles

17. Trigonometric formulas
Sums and differences of ratios

18. Trigonometric formulas
Products of ratios

19. Remember an identity is an equation that is true for all defined values of a variable.
We are going to use the identities that we have already established to "prove" or establish other identities. Let's summarize the basic identities we have.

20. RECIPROCAL IDENTITIES
QUOTIENT IDENTITIES
PYTHAGOREAN IDENTITIES
EVEN-ODD IDENTITIES

21. Even and Odd Functions
An odd function is one where f(-x) = -f(x) and an even function is one where f(-x) = f(x).

22. Let's sub in here using reciprocal identity
Establish the following identity:
Let's sub in here using reciprocal identity
We are done! We've shown the LHS equals the RHS
We often use the Pythagorean Identities solved for either sin2 or cos2.
sin2 + cos2 = 1 solved for sin2 is sin2 = 1 - cos2 which is our left-hand side so we can substitute.
In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match.

23. Let's sub in here using reciprocal identity and quotient identity
Establish the following identity:
Let's sub in here using reciprocal identity and quotient identity
We worked on LHS and then RHS but never moved things across the = sign
FOIL denominator
combine fractions
Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom

24. Get common denominators
Hints for Establishing Identities
Get common denominators
If you have squared functions look for Pythagorean Identities
Work on the more complex side first
If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity
When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities
Have fun with these---it's like a puzzle, can you use identities and algebra to get them to match!

28. See what you get

34. Etc.

35. Change everything on both sides to
Suggestions
Start with the more complicated side
Try substituting basic identities (changing all functions to be in terms of sine and cosine may make things easier)
Try algebra: factor, multiply, add, simplify, split up fractions
If you’re really stuck make sure to:
Change everything on both sides to
sine and cosine.

36. All Students Take Calculus.
Quad I
Quad II
cos(A)>0
sin(A)>0
tan(A)>0
sec(A)>0
csc(A)>0
cot(A)>0
cos(A)<0
sin(A)>0
tan(A)<0
sec(A)<0
csc(A)>0
cot(A)<0
cos(A)<0
sin(A)<0
tan(A)>0
sec(A)<0
csc(A)<0
cot(A)>0
cos(A)>0
sin(A)<0
tan(A)<0
sec(A)>0
csc(A)<0
cot(A)<0
Quad IV
Quad III

37. Reference Angles Quad I Quad II θ’ = 180° – θ θ’ = θ θ’ = π – θ
θ’ = θ + 180°
θ’ = 360° – θ
θ’ = θ + π
θ’ = 2π – θ
Quad III
Quad IV

39. Trigonometric Identities Summation & Difference Formulas

40. Trigonometric Identities Double Angle Formulas

41. Trigonometric Identities Half Angle Formulas
The quadrant of
determines the sign.

42. Work with identities to simplify an expression with the appropriate trigonometric substitutions.
For example, if x = 3 tan , then

43. Expressing One Function in Terms of Another
Example Express cos x in terms of tan x.
Solution Since sec x is related to both tan x and cos x
by identities, start with tan² x + 1 = sec² x.
Choose + or – sign, depending on the quadrant of x.

44. Rewriting an Expression in Terms of Sine and Cosine
Example Write tan  + cot  in terms of sin  and
cos .
Solution

45. 9.1 Verifying Identities Learn the fundamental identities.
Try to rewrite the more complicated side of the equation so that it is identical to the simpler side.
It is often helpful to express all functions in terms of sine and cosine and then simplify the result.
Usually, any factoring or indicated algebraic operations should be performed. For example,
As you select substitutions, keep in mind the side you are not changing, because it represents your goal.
If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin² x, which could be replaced with cos² x.

46. Verifying an Identity ( Working with One Side)
Example Verify that the following equation is an identity.
cot x + 1 = csc x(cos x + sin x)
Analytic Solution Since the side on the right is more
complicated, we work with it.
Original identity
Distributive property
The given equation is an identity because the left side equals the right side.

47. Verifying an Identity
Example Verify that the following equation is an identity.
Solution

48. Verifying an Identity ( Working with Both Sides)
Example Verify that the following equation is an identity.
Solution

49. Verifying an Identity ( Working with Both Sides)
Now work on the right side of the original equation.
We have shown that

50. Using the unit circle r

51. Draw the graph of
On the same axis:
Add the graphs together to make the blue line shown below.
The graph of

52. Two more identities starting with
Divide by
Divide by

53. Prove the following:
use the identity

54. Try these with no hints

55. The unit circle

56. Complete the radian row of this table using exact numbers30 45 60 90
120
135
150
180
210
225
240
270
300
315
330
360

57. Radians should be: 30 45 60 90
120
135
150
180
210
225
240
270
300
315
330
360

58. The Unit Circle 1 Consider the circle shown opposite.y
Make a right angled triangle.
An expression for x:
(x,y) 1 y x x
An expression for y:
And now use tan:

59. The Unit Circle - the second quadrant
Make an expression for x: 1
Make an expression for y: y -x
Make an expression for tan:
What about the third and fourth quadrants?

60. Trig. ratios in the 1st quadrant (1)
1. Find the missing side, giving your answer as a surd.
2. Find an expression for sin .
3. Find an expression for cos .
4. Find an expression for tan .
5. Find the value of the angle, using any of 2-4.
6. Fill in the appropriate column of your table 2 1

61. Trig. ratios in the 1st quadrant (2)
1. Find the missing side, giving your answer as a surd.
2. Find an expression for sin .
3. Find an expression for cos .
4. Find an expression for tan .
5. Find the value of the angle, using any of 2-4.
6. Fill in the appropriate column of your table 1 1

62. Trig. ratios in the 1st quadrant (3)
1. Find the missing side, giving your answer as a surd.
2. Find an expression for sin .
3. Find an expression for cos .
4. Find an expression for tan .
5. Find the value of the angle, using any of 2-4.
6. Fill in the appropriate column of your table 2 1

63. Trig. ratios in the 2nd quadrant
1. Find the missing side, giving your answer as a surd.
2. Find an expression for sin .
3. Find an expression for cos .
4. Find an expression for tan .
5. Find the value of the angle, using any of 2-4.
6. Fill in the appropriate column of your table 2 -1
Use your answers from previous slides to fill in all the angles trig. values in the second quadrant.

64. Trig. ratios in the 3rd quadrant
1. Find the missing side, giving your answer as a surd.
2. Find an expression for sin .
3. Find an expression for cos .
4. Find an expression for tan .
5. Find the value of the angle, using any of 2-4.
6. Fill in the appropriate column of your table -1 -1
Use your answers from previous slides to fill in all the angles trig. values in the third quadrant.

65. Trig. ratios in the 4th quadrant
1. Find the missing side, giving your answer as a surd.
2. Find an expression for sin .
3. Find an expression for cos .
4. Find an expression for tan .
5. Find the value of the angle, using any of 2-4.
6. Fill in the appropriate column of your table -1 2
Use your answers from previous slides to fill in all the angles trig. values in the second quadrant.

66. The completed table 30 45 60 90
120
135
150
180 1 -1 -
210
225
240
270
300
315
330
360 -1 1 -