Analytic Trigonometry and Trig-Formulas

1. Analytic Trigonometry

2. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().

(1,0)
The unit circle x2 + y2 = 1
(x, y)

3. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
The unit circle x2 + y2 = 1
(x, y)

4. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
The unit circle x2 + y2 = 1
(x, y)

5. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
cot()
The unit circle x2 + y2 = 1
(x, y)

6. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
cot()
The following reciprocal
functions are useful in
science and engineering:
csc() = 1/sin() = 1/y,
the “cosecant of ".
sec() = 1/cos() = 1/x,
the "secant of "
(x, y)
The unit circle x2 + y2 = 1

7. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
cot()
The following reciprocal
functions are useful in
science and engineering:
csc() = 1/sin() = 1/y,
the “cosecant of ".
sec() = 1/cos() = 1/x,
the "secant of "
Since y2 + x2 = 1, we’ve sin2() + cos2() = 1
or s()2 + c()2 = 1 or simply s2 + c2 = 1 all angles .
This is the most important identity in trigonometry..
(x, y)
The unit circle x2 + y2 = 1

8. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
cot()
The following reciprocal
functions are useful in
science and engineering:
csc() = 1/sin() = 1/y,
the “cosecant of ".
sec() = 1/cos() = 1/x,
the "secant of "
Since y2 + x2 = 1, we’ve sin2() + cos2() = 1
or s()2 + c()2 = 1 or simply s2 + c2 = 1 all angle .
This is the most important identity in trigonometry.
(x, y)
The unit circle x2 + y2 = 1

9. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().

10. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
If we divide the identity s2 + c2 = 1 by c2, we obtain
(s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().

11. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
If we divide the identity s2 + c2 = 1 by c2, we obtain
(s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().
The Trig–HexagramThe hexagram shown here
is a useful visual tool for
recalling various relations.

12. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
If we divide the identity s2 + c2 = 1 by c2, we obtain
(s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().
The Trig–HexagramThe hexagram shown here
is a useful visual tool for
recalling various relations.
The right side of the hexagram
is the “co”–side listing all the
co–functions.
The “co”–side

13. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
If we divide the identity s2 + c2 = 1 by c2, we obtain
(s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().
The Trig–HexagramThe hexagram shown here
is a useful visual tool for
recalling various relations.
The right side of the hexagram
is the “co”–side listing all the
co–functions.
The following three groups
of formulas extracted from this
hexagram are referred to as
the fundamental trig–identities.
The “co”–side

14. Analytic Trigonometry
The Division Relations

15. Analytic Trigonometry
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II

16. Analytic Trigonometry
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) =
sec(A) =
sin(A)cot(A) =

17. I
II III
Analytic Trigonometry
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) =
sin(A)cot(A) =
÷

18. I II
III
Analytic Trigonometry
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) =
÷

19. I II
III
Analytic Trigonometry
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)

20. Analytic Trigonometry
The Reciprocal Relations
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)

21. Analytic Trigonometry
The Reciprocal Relations
Start from any function, going across diagonally,
we always have I = 1 / III.
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II.
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)

22. I
III
Analytic Trigonometry
The Reciprocal Relations
Start from any function, going across diagonally,
we always have I = 1 / III.
Example B: sec(A) = 1/cos(A),
÷II
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II.
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)

23. IIII
Analytic Trigonometry
The Reciprocal Relations
Start from any function, going across diagonally,
we always have I = 1 / III.
Example B. sec(A) = 1/cos(A), cot(A) = 1/tan(A)
The Division Relations
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II.
Example A.
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)
÷ II

24. Square–Sum Relations
Analytic Trigonometry

25. Square–Sum Relations
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
Analytic Trigonometry

26. Square–Sum Relations
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
sin2(A) + cos2(A) = 1
Analytic Trigonometry

27. Square–Sum Relations
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
sin2(A) + cos2(A) = 1
tan2(A) + 1 = sec2(A)
Analytic Trigonometry

28. Square–Sum Relations
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
sin2(A) + cos2(A) = 1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Analytic Trigonometry

29. Square–Sum Relations
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
sin2(A) + cos2(A) = 1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Analytic Trigonometry
Identities from this hexagram in the above manner
are called the fundamental Identities.
We assume these identities from here on and list the
most important ones below.

30. Fundamental Identities
Division Relations
tan(A)=S/C
cot(A)=C/S

31. Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
C = cos(A)
S = sin(A)
T = tan(A)

32. Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square–Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
C = cos(A)
S = sin(A)
T = tan(A)

33. Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square–Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Two angles are complementary if their sum is 90o.
A
B
A and B are complementary
C = cos(A)
S = sin(A)
T = tan(A)

34. Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square–Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Two angles are complementary if their sum is 90o.
Let A and B = 90 – A be complementary angles,
we have the following co–relations.
A
B
A and B are complementary
C = cos(A)
S = sin(A)
T = tan(A)

35. Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square–Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Two angles are complementary if their sum is 90o.
Let A and B = 90 – A be complementary angles,
we have the following co–relations.
S(A) = C(B) = C(90 – A)
T(A) = cot(B) = cot(90 – A)
sec(A) = csc(B) = csc(90 – A)
A
B
A and B are complementary
C = cos(A)
S = sin(A)
T = tan(A)

36. cos(–A) = cos(A), sin(–A) = – sin(A)
The Negative Angle Relations
Fundamental Identities
1
A
–A
1
A
–A
sin(A)
sin(–A)

37. cos(–A) = cos(A), sin(–A) = – sin(A)
The Negative Angle Relations
Fundamental Identities
Recall that f(x) is even if and only if f(–x) = f(x),
and that f(x) is odd if and only if f(–x) = –f(x).
1
A
–A
1
A
–A
sin(A)
sin(–A)

38. cos(–A) = cos(A), sin(–A) = – sin(A)
The Negative Angle Relations
Fundamental Identities
1
A
–A
1
A
–A
Therefore cosine is an even function and
sine is an odd function.
Recall that f(x) is even if and only if f(–x) = f(x),
and that f(x) is odd if and only if f(–x) = –f(x).
sin(A)
sin(–A)

39. cos(–A) = cos(A), sin(–A) = – sin(A)
The Negative Angle Relations
Fundamental Identities
1
A
–A
1
A
–A
Therefore cosine is an even function and
sine is an odd function.
Since tangent and cotangent are quotients
of sine and cosine, so they are also odd.
Recall that f(x) is even if and only if f(–x) = f(x),
and that f(x) is odd if and only if f(–x) = –f(x).
sin(A)
sin(–A)

40. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.

41. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))

42. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))

43. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))
c. cot2(A) – csc2(A) = –1
= (cot(A) – csc(A))(cot(A) + csc(A))

44. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))
c. cot2(A) – csc2(A) = –1
= (cot(A) – csc(A))(cot(A) + csc(A))
Example D. Simplify (sec(x) – 1)(sec(x) + 1)
And express the answer in sine and cosine.

45. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))
c. cot2(A) – csc2(A) = –1
= (cot(A) – csc(A))(cot(A) + csc(A))
Example D. Simplify (sec(x) – 1)(sec(x) + 1)
And express the answer in sine and cosine.
(sec(x) – 1)(sec(x) + 1) = sec2(x) – 1

46. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))
c. cot2(A) – csc2(A) = –1
= (cot(A) – csc(A))(cot(A) + csc(A))
Example D. Simplify (sec(x) – 1)(sec(x) + 1)
And express the answer in sine and cosine.
(sec(x) – 1)(sec(x) + 1) = sec2(x) – 1 = T2(x) = S2(x)
C2(x)

47. Notes on Square–Sum Identities
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.

48. Notes on Square–Sum Identities
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.

49. Notes on Square–Sum Identities
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.

50. Notes on Square–Sum Identities
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.

51. Notes on Square–Sum Identities
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.

52. Notes on Square–Sum Identities
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
or that S = 1, so x = π/2.
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.

53. Notes on Square–Sum Identities
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
or that S = 1, so x = π/2.
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.

54. Notes on Square–Sum Identities
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
or that S = 1, so x = π/2.
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
By adding 2nπ where n is an integer, to the solutions
from a. we obtain all the solutions for x.

55. Notes on Square–Sum Identities
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
or that S = 1, so x = π/2.
Example E. Solve the equation
S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
By adding 2nπ where n is an integer, to the solutions
from a. we obtain all the solutions for x. Here is a list:
{π/6 or π/2 or 5π/6 + 2nπ, where n is an integer}.