1. Fundamental Identities
Math-2 Honors Lesson 12.2
Fundamental Identities

2. What you’ll learn about
Identities
Basic Trigonometric Identities
Pythagorean Identities
Negative Angle Identities
Even/Odd Identities
Simplifying Trigonometric Expressions
Solving Trigonometric Equations
… and why
Identities are important when working with trigonometric
functions in calculus.

3. Trigonometric Functions
SOHCAHTOA
“Some old horse
caught another horse
taking oats away.”
SOH
CAH
TOA

4. Trigonometric Functions
Cosecant ratio

5. Trigonometric Functions
Secant ratio

6. Trigonometric Functions
Cotangent ratio
Shot your cow: “Sha – Cho – Cao”

7. Trigonometric Functions
Shot your cow: “Sha – Cho – Cao”
Unfortunately, ‘s’ doesn’t match up with ‘s’ (or ‘c’ with ‘c’)

8. SHA-CHO-CAO
SOHCAHTOA
“shot your cow”
SHA
SOH
CAH
CHO
TOA
CAO

9. What is an Identity? 2(x – 3) = 2x – 6
This is an equivalent expression.
It is true for all real numbers.
2(x – 3) = 2x – 6
Like the equation above, it is a
true statement. BUT, it is only true
if x is in the domain of the expression
on the right AND left side.
When x = -1, it is not true.
Identity: an equation that is true for all values that are in
the domain of both sides of the equation.

10. Identity: an equation that is true for all values that are in
the domain of both sides of the equation.
Is it an Identity? 1. 2.
NOT an identity
Domain: left side: x ≠ 2
right side: all real #’s
NOT an identity

11. Basic Trigonometric Identities

12. Find the Product

13. Find the Product/quotient3. 6. 4. 5.

14. Using the UNIT CIRCLE y x r = 1 Sine of the angle
(x, y) y
r = 1 x
Sine of the angle
is the vertical distance from the x-axis to the point on the circle.

15. Using the UNIT CIRCLE y x r = 1 Cosine of the angle
(x, y) y
r = 1 x
Cosine of the angle
is the horizontal distance from the y-axis to the point on the circle.

16. Using the UNIT CIRCLE y x r = 1 The ordered pair (x, y) can
be re-written as:
(x, y) y
r = 1 x

17. y x Using the UNIT CIRCLE r = 1 Using Pythagorean Theorem:
We usually write
as: x
This give us our 1st
“Pythagorean” identity.

18. Pythagorean Identities
Divide both sides of the equation by
This give us our 2nd
“Pythagorean” identity.
Your turn:
3. Divide the first identity by and simplify to
find the 3rd Pythagorean Identity.

19. Solving equations
To solve an equation we would use properties of equality
to “isolate the variable” on one side of the equal sign.
When using identities to solve equations we use
substitution.

20. Using Identities Find sin if cos Which of these identities will help?
Substitution step.

21. Your Turn:
Given:
Given:
7. Find:
8. Find:

22. Using Identities Find sin and cos if tan = 4
Which of these identities will help?
Substitution step.

23. Using Identities Find sin and cos if tan = 4
Which of these identities will help?
Substitution step.

24. Your Turn:
Given: sec(x) = 4,
9. Find: tan(x)
Find: cot(x)

25. function of angle A = “cofunction” of angle B.

26. Negative Angle Identities (“odd-even” identities)
Sin θ = y coordinate of
the point on the circle.
sin(-θ) = -sin(θ)
(x, y)
“the y coord. of point
through which (-θ) passes
is the negative of the
y-coord of the point
through which (θ) passes. θ -θ
(x, -y)

27. Even-Odd Identities cos θ = x coordinate of the point on the circle.
(x, y)
Cos (-θ) = cos (θ)
“the x coord. of point
through which (-θ) passes
is the same as the
x-coord of the point
through which (θ) passes. θ -θ
(x, -y)

28. Even-Odd Identities (x, y) Sin (-θ) = -sin (θ) csc (-θ) = - csc (θ) θ
Cos (-θ) = cos (θ)
sec (-θ) = sec (θ)
(x, -y)

29. Even-Odd Identities Sin (-θ) = -sin (θ) Cos (-θ) = cos (θ) (x, y) θ -θ
tan (-θ) = - tan (θ)
cot (-θ) = - cot (θ)
(x, -y)

30. Even-Odd Identities Find: The book uses ‘x’ instead of ‘θ’
for the angle variable.
Find: x -x

31. Your Turn:
11.
12.

32. Simplifying by Factoring and Using Identities
Try converting tan, cot, sec, csc into functions of sin and cos
From the properties
of exponents:
From the inverse
Property of multiplication:
From the Pythagorean
Property

33. Your turn:
13. Simplify:
Pythagorean Identity

34. Simplifying expressions with Identities
Anytime you see
refer to the Pythagorean Identities.
Use substitution

35. Your turn: use identities to simplify
13.

36. 14.
Convert into “sines” and “cosines”
Need common denominator!

37. 15.
Convert into “sines” and “cosines”