Download trigonometric identities

TRIGONOMETRY
First, an experiment:
1. Draw a circle with an origin of (0, 0)
2. Mark and label a point in each of the four quadrants
(A, B, C, D)
3. Find sin, cos, and tan for each point
Ex. Sin A =
Cos A =
Tan A =
Q2
Q1
Q3
Q4
TRIGONOMETRIC RATIOS
And now…definitions
Standard position - an angle whose vertex lies at the
origin and whose initial arm lies on the
positive x-axis
Principal angle
- the counter clockwise angle between
the initial arm and terminal arm
Related acute angle - the acute angle between the terminal
arm and the x-axis
Negative angle
- an angle measured clockwise from the
positive x-axis
Primary Ratios
Ex1 Find the missing side
a)
b)
Ex2 Find the missing angle
a)
b)
Reciprocal Ratios
Ex1 State the six trig ratios for each.
a)
13
5
12
b) tanθ =
3
4
Ex2 Determine the angle to the nearest hundredth
a) cot θ = 3.2404
b) csc θ = 1.2711
c) sec θ = 1.4526
SPECIAL ANGLES
Sin 45° =
Cos 45° =
Tan 45° =
Sin 30° =
sin 60° =
Cos 30° =
cos 60° =
Tan 30° =
tan 60° =
Ex1 Determine the exact value
a) (2sin45º)(sin45º)
b) 1 -
sin 45
cos 45
c) (sin30º)(tan60º) – cos30º
ANGLES > 90°
Let’s look back at our experiment
Q1 - sin
Q2 - sin
- cos
- cos
- tan
- tan
Q3 - sin
Q4 - sin
- cos
- cos
- tan
- tan
Cast Rule
S
A
T
C
C – Cosine
A – All
S – Sine
T – Tangent
To find angles with the same ratios:
- find principal angle
- use CAST rule to determine quadrant of second
angle
- use related acute angle to determine second angle
Ex1 Determine the angle(s) that satisfy each for 0 ≤ θ ≤ 360
a)
sin  
1
3
b)
cot  
4
5
c) (-8, 3)
Date
1
2
3
4
D1
Thursday
July 02
1
2
3
4
1
D2
Friday
July 03
2
3
4
1
2
3
4
1
Monday
July06
2
3
4
1
2
3
4
1
D3
D4
Tuesday
July07
D5
Wednesday
July08
Thursday
D6
July 09
D7
STRAND: CHARACTERISTICS OF FUNCTIONS
Unit 1- Characteristics of functions
Relations and Functions
Function Notation
Properties of parent functions
Domain and range of a function
Ass
Inverse function
Transformations of functions
Transformations of functions
Review
Unit 1-Test-Students will:
E.1. demonstrate an understanding of functions, their representations, and their inverses, and make
connections between the algebraic and graphical representations of functions using transformations
Unit 2- Quadratic Functions
Properties of Quadratic Functions
Max/Min values of quadratics
Inverse of a quadratic
Solving Quadratics
Solving Quadratics
Linear/ Quadratic Systems
Review
Unit 2- Test –Students will:
E.2-determine the zeros, and the maximum or minimum of a quadratic function, and solve problems
involving quadratic functions, including problems arising from real-world applications;
Unit 3- Equivalent Algebraic Expressions
Operations with Radicals
Operations with polynomials
Factoring polynomials
Simplifying Rational Expressions
Multiplying and Dividing Rational expressions
Adding and Subtracting Rational Expressions
Review
Unit 3- Test-Students will:
E.3 demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical,
and rational expressions.
P.4
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P.7
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Trig Ratios in the Cartesian Plane
For any point P(x, y) in the Cartesian plane, the trig ratios
for angles in standard positin can be represented in terms of
x, y, and r.
Given a circle with radius r
Ex2 What will be the new point if (5, 0) is rotated about the
origin by 238º?
p. 304 #1 - 13
TRIGONOMETRIC IDENTITIES
Identity - a mathematical statement that is true for all
values of the given variables.
csc  
1
sin 
tan  
sin 
cos 
sin 2   cos 2   1
sec  
1
cos 
cot  
cos 
sin 
1  tan 2   sec 2 
cot  
1
tan 
1  cot 2   csc 2 
sin 
cos 
Prove
tan  
Prove
sin 2   cos 2   1
Prove 1  tan
2
  sec 2 
Ex1 Prove the following identities
a) 1  cos
b)
2
  sin  cos  tan 
sin 
1  cos 

1  cos 
sin 
Pg.310#1ad, 5,7,8,12
SINE LAW
- 2 sides and opposite angle (SSA)
- 2 angles and 1 side (AAS)
a
b
c


sin A sin B sin C
sin A sin B sin C


a
b
c
Ex1 Determine angle B
Ex2 Determine side a
The Ambiguous Case (SSA)
- a < h, no solutions
- a = h, 1 solution
- a > b, 1 solution
- h < a < b, 2 solutions
Ex1 Determine the number of triangles possible
a) a = 7.2, b = 9.3, A = 35º
b) a = 7.3, b = 14. 6, A = 30º
c) a = 1.3, b =2.8, A = 33º
d) COSINE LAW
- 3 sides (SSS)
- 2 sides and contained angle (SAS)
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
Ex1 Determine the unknown side
Ex2 Determine the unknown angle
3D WORD PROBLEMS
Angle of elevation - angle up from a horizontal
Angle of depression- angle down from a horizontal
Bearing
Pg.318 #1,2,5,9,15
pg.325#4ac,7,8
Pg.332#3,7,8
- angle clockwise from N