This document contains sections from a trigonometry textbook chapter on analytic trigonometry and trigonometric identities. It includes explanations of trigonometric identities and examples of simplifying trigonometric expressions using identities. It also contains a biblical verse and encourages students to study examples, do their homework, and ask questions in class.
1. Chapter 7
Analytic Trigonometry
Matthew 19:25-26
When the disciples heard this, they were greatly
astonished and asked, "Who then can be saved?"
Jesus looked at them and said, "With man this is
impossible, but with God all things are possible."
2. Chapter 7
Analytic Trigonometry
Much of this chapter will be new topics for you.
Read your textbook! Study the examples!!
Keep current with your homework!!!
Matthew 19:25-26
When the disciples heard this, they were greatly
astonished and asked, "Who then can be saved?"
Jesus looked at them and said, "With man this is
impossible, but with God all things are possible."
5. 7.1 Trigonometric Identities
An identity is an equation that is true for all values
of the variable in the domain.
An equation will be true for one or more, but not
all, values of the variable in the domain.
6. 7.1 Trigonometric Identities
An identity is an equation that is true for all values
of the variable in the domain.
An equation will be true for one or more, but not
all, values of the variable in the domain.
Equation : 4x − 3 = 5
7. 7.1 Trigonometric Identities
An identity is an equation that is true for all values
of the variable in the domain.
An equation will be true for one or more, but not
all, values of the variable in the domain.
Equation : 4x − 3 = 5
2x + 14
Identity : x+7=
2
10. Fundamental Trig Identities
(open books to page 528 ... look at the box)
Reciprocal: you already know these
Pythagorean: hexagon on your Unit Circle
11. Fundamental Trig Identities
(open books to page 528 ... look at the box)
Reciprocal: you already know these
Pythagorean: hexagon on your Unit Circle
Even-Odd: on your help sheet
12. Fundamental Trig Identities
(open books to page 528 ... look at the box)
Reciprocal: you already know these
Pythagorean: hexagon on your Unit Circle
Even-Odd: on your help sheet
Cofunction: on your help sheet
15. Simplifying Trig Expressions
Use identities and other math operations to rewrite
a trig expression
We use this technique a lot when proving trig
identities
24. Simplify
1− cos x sin x
+
sin x 1− cos x
1− cos x 1− cos x sin x sin x
⋅ + ⋅
sin x 1− cos x 1− cos x sin x
25. Simplify
1− cos x sin x
+
sin x 1− cos x
1− cos x 1− cos x sin x sin x
⋅ + ⋅
sin x 1− cos x 1− cos x sin x
2 2
(1− cos x ) + sin x
sin x (1− cos x )
26. Simplify
1− cos x sin x
+
sin x 1− cos x
1− cos x 1− cos x sin x sin x
⋅ + ⋅
sin x 1− cos x 1− cos x sin x
2 2
(1− cos x ) + sin x
sin x (1− cos x )
1− 2 cos x + cos 2 x + sin 2 x
sin x (1− cos x )
27. Simplify
1− cos x sin x
+
sin x 1− cos x
1− cos x 1− cos x sin x sin x
⋅ + ⋅
sin x 1− cos x 1− cos x sin x
2 2
(1− cos x ) + sin x
sin x (1− cos x )
1− 2 cos x + cos 2 x + sin 2 x
sin x (1− cos x )
2 − 2 cos x
sin x (1− cos x )
28. Simplify
1− cos x sin x
+
sin x 1− cos x
1− cos x 1− cos x sin x sin x
⋅ + ⋅
sin x 1− cos x 1− cos x sin x
2
(1− cos x ) + sin x2
2 (1− cos x )
sin x (1− cos x ) sin x (1− cos x )
1− 2 cos x + cos 2 x + sin 2 x
sin x (1− cos x )
2 − 2 cos x
sin x (1− cos x )
29. Simplify
1− cos x sin x
+
sin x 1− cos x
1− cos x 1− cos x sin x sin x
⋅ + ⋅
sin x 1− cos x 1− cos x sin x
2
(1− cos x ) + sin x2
2 (1− cos x )
sin x (1− cos x ) sin x (1− cos x )
1− 2 cos x + cos 2 x + sin 2 x 2 csc x
sin x (1− cos x )
2 − 2 cos x
sin x (1− cos x )
31. Simplify
csc x cot x
−
sin x tan x
1 cos x
sin x − sin x
sin x sin x
1 cos x
32. Simplify
csc x cot x
−
sin x tan x
1 cos x
sin x − sin x
sin x sin x
1 cos x
1 cos 2 x
2
− 2
sin x sin x
33. Simplify
csc x cot x
−
sin x tan x 1− cos 2 x
2
1 cos x sin x
sin x − sin x
sin x sin x
1 cos x
1 cos 2 x
2
− 2
sin x sin x
34. Simplify
csc x cot x
−
sin x tan x 1− cos 2 x
2
1 cos x sin x
sin x − sin x 2
sin x
sin x sin x 2
sin x
1 cos x
1 cos 2 x
2
− 2
sin x sin x
35. Simplify
csc x cot x
−
sin x tan x 1− cos 2 x
2
1 cos x sin x
sin x − sin x 2
sin x
sin x sin x 2
sin x
1 cos x
2 1
1 cos x
2
− 2
sin x sin x
36. For the next few days ... I’ll draw names “out of
the hat”. Those people will be chosen to put a
homework problem on the board. When I draw a
name, it will be for a particular problem ...
37. HW #1
For every pass I caught in a game, I caught a
thousand in practice.
Don Hutson
Notas do Editor
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1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n