Pre-Calculus
Learning Targets
 Review Reciprocal Trig Relationships
 Explain the relationship of trig functions with positive
and negative angles
 Explain the Pythagorean trig relationships
 Explain the Cofunction trig relationships
 Apply various trig relationships to simplify expressions
Review of Reciprocal Trig
Relationships
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Example 1: Simplifying Expressions
 Simplify the following Expressions
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Part 1:
Trig Relationships with Negative & Positive Angles
 Let’s first take a look at a positive and negative angle
on the unit circle
Part 1:
Trig Relationships with Negative & Positive Angles
 Let’s take a look at
What does this equal
according to our picture?
 What about
What does this equal
according to our picture?
 What can we say about the relationship between
Part 1:
Trig Relationships With Negative and Positive Angles
 We just proved that sin (-θ) = - sin θ
 What do you think the relationship between cos (- θ)
and cos θ is?
 cos (- θ) = cos θ
 What about the relationship between tan (- θ) and
tan θ?
 tan (- θ) = - tan θ
Part 1:
Trig Relationships With Negative and Positive Angles
 Let’s look at csc (- θ) and csc θ. What is the
relationship?
 csc (- θ) = - csc θ
 What about the relationship between sec (- θ) and
sec θ?
 sec (- θ) = sec θ
 What about the relationship between cot (- θ) and
cot θ?
 cot (- θ) = - cot θ
Examples: Practice Simplifying
 Write the equivalent trig function with a positive angle
 Sin (-π/2)
 Cos (-π/3)
 Cot (-3π/4)
Part 2:
Pythagorean Trig Relationships
 Let’s take a look at the unit circle.
 Using the Pythagorean Theorem, how can you relate
all three sides of the triangle?
 sin2θ + cos2θ = 1
 This is one of the Pythagorean
Trig Relationships
Part 2:
Pythagorean Trig Relationships
 Starting with sin2θ + cos2θ = 1, how can you manipulate it
to get other following Pythagorean Trig Relationships?
 1 + tan2θ = sec2θ
 Divide both sides by cos2θ
 1 + cot2θ = csc2θ
 Divide both sides by sin2θ
 These are the final 2 of the 3 Pythagorean Trig
Relationships
Examples: Simplifying Expressions
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Part 3:
Cofunction Trig Relationships
 Sine & Cosine, Tangent & Cotangent, Secant &
Cosecant are all Cofunctions.
 Trig Cofunctions have the following relationship
 The relationships still hold if the angle is in radians
(π/2)
Examples: Simplifying Expressions
 Simplify the following
 tan (90° – A) =
 Cos (π/2 – x) =
Tips to help simplify expressions
 There are 4 different categories of trig relationships
which each have different key components to look for
 Reciprocal Relationships
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Most commonly used in some type of format similar to
 cot y · sin y
 manipulating a fraction with trig functions
 Usually the functions aren’t squared when they are in this
format
 Negative/Positive Angle Relationships
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Similar to the example problems previously in this powerpoint
tan (-45°)
Tips to help simplify expressions
 There are 4 different categories of trig relationships
which each have different key components to look for
 Cofunction Relationships
 Similar to the example problems previously in this powerpoint
 cos (90° – A)
 Pythagorean Relationships (MOST COMMON/CHALLENGING!)
 Includes exponents to the second degree
 Includes expanding two binomials
 Addition and subtraction of fractions
 May need to factor out a trig function before simplifying
 Or some type of variation of the previous
Tips to help simplify expressions
 Though most of the problems are separated into their
respective categories, you may find yourself having to
combine multiple relationships to fully simplify an
expression.
 Maybe you’ll start with Pythagorean relationships, then
to fully simplify you may use Reciprocal relationships.
 In most cases, fully simplifying an expression will leave
the expression with only one term
Homework
 Textbook pg 321: #1, 12, 13, 19