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Fundamental Identities

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Presentation on theme: "Fundamental Identities"— Presentation transcript:

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/quotient
3. 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 (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 (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”


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