brought to you by tutorial services – the math center trigonometric identities
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Brought to you by Brought to you by Tutorial Services – The Math CenterTutorial Services – The Math Center
Trigonometric IdentitiesTrigonometric Identities
In this workshop we will:In this workshop we will:
Look at basic Identities.Look at basic Identities. How other identities can be derived from How other identities can be derived from
basic identities.basic identities. Look at Sum and Difference Identities.Look at Sum and Difference Identities. Look at Double-Angle and Half-Angle Look at Double-Angle and Half-Angle
Identities.Identities. Look at Product and Sum Identities.Look at Product and Sum Identities. How to develop a strategy for solving How to develop a strategy for solving
trigonometric identities.trigonometric identities.
Basic IdentitiesBasic IdentitiesEvery trigonometric function is related to the other
because they are all defined in terms of the coordinates on a unit circle.
Identities from the Definitions
cos
sintan
sin
coscot
cos
1sec
sin
1csc
Basic IdentitiesBasic Identities
From the basic Identities we can define the Reciprocal Identities
Reciprocal Identities
csc
1sin
sec
1cos
cot
1tan
sin
1csc
cos
1sec
tan
1cot
Basic IdentitiesBasic Identities
The Pythagorean Identities can be derived from the fundamental identity
Pythagorean Identities
1cossin 22 xx
1cossin 22 xx
xx 22 csc1cot
xx 22 sec1tan
Sum and Difference IdentitiesSum and Difference Identities
These identities are used in solving equations and in simplifying expressions.
Sum or Difference (Sines, Cosines)
sinsincoscos)cos(
sinsincoscos)cos(
sincoscossin)sin(
sincoscossin)sin(
Sum and Difference IdentitiesSum and Difference Identities
These identities are used in solving equations and in simplifying expressions.
Sum or Difference (Tangents)
tantan1
tantan)tan(
tantan1
tantan)tan(
Double-Angle and Half-Angle IdentitiesDouble-Angle and Half-Angle Identities
xxx cossin22sin xxx 22 sin211cos22cos
The double-angle and half-angle identities are special cases of those identities.
Double-Angle Identities
x
xx
2tan1
tan22tan
Double-Angle and Half-Angle IdentitiesDouble-Angle and Half-Angle Identities
The double-angle and half-angle identities are special cases of those identities.
Half-Angle Identities
2
cos1
2sin
xx
2
cos1
2cos
xx
x
x
x
xx
cos1
sin
sin
cos1
2tan
Product and Sum IdentitiesProduct and Sum Identities
These identities are used to solve certain problems, but not used as often.
Product-to-Sum Identities
)]sin()[sin(2
1cossin BABABA
)]cos()[cos(2
1sinsin BABABA
)]sin()[sin(2
1sincos BABABA
)]cos()[cos(2
1coscos BABABA
Product and Sum IdentitiesProduct and Sum Identities
These identities are used to solve certain problems, but not used as often.
Sum-to-Product Identities
2
cos2
sin2sinsinyxyx
yx
2
sin2
cos2sinsinyxyx
yx
2
cos2
cos2coscosyxyx
yx
2
sin2
sin2coscosyxyx
yx
Developing StrategyDeveloping Strategy
Verifying identities takes practice! The goal is to prove that both sides are equal to one
another.
1. You may work with one or both sides of the equation.2. Rewrite the expressions in terms of sines and cosines only.3. Other algebraic methods can be used such as factoring,
finding the LCD or cross-multiplying.
Verify that the following is an identity:
xxxx 2sectansinsec1
Developing StrategyDeveloping Strategy
xxxx 2sectansinsec1
xx
xx
x2sec
cos
sinsin
cos
11
xx
x 22
2
seccos
sin1
Solution:
This is now verified by the Pythagorean Identity
xx 22 sec1tan
Developing StrategyDeveloping Strategy
Tip: Basic cross-multiplication can simplify your verification!
Now try some on your own.Verify that the following is an identity:
cos
sin1
sin1
cos
Developing StrategyDeveloping Strategy
Tip: Stick to sines and cosines and work with both sides of the equation. Don’t forget your rules for basic math!
Verify that the following is an identity:
xx
xx 2cotsin
sincsc
Trigonometric IdentitiesTrigonometric Identities Links Links
•Trigonometric Identities HandoutTrigonometric Identities Handout•Trigonometric Formulas HandoutTrigonometric Formulas Handout•Equation of a Circle HandoutEquation of a Circle Handout•Trigonometric Substitution QuizTrigonometric Substitution Quiz•Trigonometric Identities QuizTrigonometric Identities Quiz