1. Analytic Trigonometry 5.1 FUNdamental Identities
Haha, FUNdamental. Get it? Fundamental…FUN…nm.

2. Our primary applications of trigonometry so far have been computational. We have not made full use of the properties of the functions to study the connections among the trigonometric functions themselves. We will now shift our emphasis more toward theory and proof, exploring where the properties of these special functions lead us, often with no immediate concern for real-world relevance at all. Hopefully in the process you will gain an appreciation for the rich and intricate tapestry of interlocking patterns that can be woven from the six basic trig functions – patterns that will take on even greater beauty later on when you can view them through the lens of calculus.

3. 5.1 Fundamental Identities
During today’s lesson you will learn:
Definition of an identity
Basic Trigonometric Identities, Pythagorean Identities, Cofunction and Even-Odd Identities
Simplify Trigonometric Expressions and Solve Trigonometric Equations

4. Which represents an identity?
1 + 1 = 2
2(x – 3) = 2x – 6
x2 + 3 = 7
(x2 – 1)/(x+1) = x – 1
What are the similarities and differences in each of the equations above?

5. What is an identity?
Identities - statements which are true for all values of the variable for which both sides of the equation are defined
In other words, an identity is an equation that is ALWAYS equal for values which are appropriate for its domain.
Domain of validity – set of values for which an equation is defined.

6. Basic Trigonometric Identities
Reciprocal Identities
Quotient Identities

7. Exploration 1 Domain of Validity
Complete the exploration on p. 445
6 min.
1st 4 min. – NO TALKING!!
Last 2 min. – You can discuss with your neighbor
Write answers on your own paper!!

8. Exploration 2 Pythagorean Theorem
Use your calculator to evaluate each expression
Write Value
What conclusions can you draw about sin2q + cos2q ?
(sin(25))2 + (cos(25))2
(sin(72))2 + (cos(72))2
(sin(90))2 + (cos(90))2
(sin(30))2 + (cos(30))2

9. Pythagorean Identities
sin2 q + cos 2 q = 1
1 + tan2 q = sec 2 q
cot 2 q + 1 = csc 2 q
How can you change the first identity into the second? The third?

10. Ex 1 Without a calculator
Find sin q & cos q if tan q = 8 & sin q > 0.

11. Exploration 3 Cofunction Identities
Type in as shown on your calculator
Compare
Evaluate
What conclusions can you draw about sin (90 - q) & cos q?
cos (90 - q) & sin q?
tan (90 - q) & cot q?
cot (90 - q) & tan q?
sin ( )
cos (65)
cos ( )
sin (71)
tan ( )
cot (68)
cot ( )
tan (47)

12. Cofunction Identities

13. Exploration 4 Odd-Even Identities
Evaluate
Compare
S or O
Which functions are even?
Which functions are odd?
sin (-25)
sin (25)
cos (-25)
cos (25)
tan (-25)
tan (25)
cot (-25)
cot (25)
csc (-25)
csc (25)
sec (-25)
sec (25)

14. Odd-Even Identities

15. Ex 2 If sin ( - p/2) = 0.73, find cos (-).
Ex 3 If cot(-) = 7.89 , find tan ( - p/2).

16. Simplifying Trigonometric Expressions
Ex 4 Use basic identities to simplify the expression. a) 1 + tan2x b) sec2 (-x) – tan2x csc2 x

17. Ex 5 Use basic identities to simplify the expression.
(sin x) (tan x + cot x)
tan x tan x
csc2 x sec2 x

18. Solving Trigonometric Equations
Ex 6 Find all values of x in the interval [0,2p) that solve 2 cos x sin x – cos x = 0.
Algebraically
Verify by graphing

19. Ex 6 Find all solutions to the equation in the interval [0,2p)
Ex 6 Find all solutions to the equation in the interval [0,2p). You do NOT need a calculator!
4 cos2 x – 4 cos x + 1 = 0
2 sin2 x + 3 sin x = 2

20. Ex 7 Use your calculator to solve
cos x = 0.75
sin2x = 0.4

21. Exiting Thoughts… Mrs. Mullen asked the class to factor 1 – sin2x.
Fundamental Identities
Exiting Thoughts…
Mrs. Mullen asked the class to factor – sin2x.
Ana wrote (1 – sin x)(1 + sin x).
Cara wrote (cos x)(cos x).
Who is correct? Explain how you made your choice.