Download Trigonometric identities

Identities and Equations
An equation such as y2 – 9y + 20 = (y – 4)(y – 5)
is an identity because the left-hand side (LHS) is
equal to the right-hand side (RHS) for whatever
value is substituted to the variable.
 Based on the example, an identity is defined as
an equation, which is true for all values in the
domain of the variable.

Identities and Equations
There are identities which involve
trigonometric functions. These identities
are called trigonometric identities.
 Trigonometric identity is an equation
that involves trigonometric functions,
which is true for all the values of θ for
which the functions are defined.

Identities and Equations
A conditional equation is an equation that is
true only for certain values of the variable.
 The equations y2 – 5y + 6 = 0 and x2 – x – 6 = 0
are both conditional equations. The first
equation is true only if y = 2 and y = 3 and the
second equation is true only if x = 3 and x = -2.

The Fundamental Identities
The Fundamental Identities
Reciprocal Identities
Reciprocal
Identities
1
csc 𝜃 =
sin 𝜃
1
sec 𝜃 =
cos 𝜃
1
cot 𝜃 =
tan 𝜃
Equivalent Forms
1
csc 𝜃
1
cos 𝜃 =
sec 𝜃
1
tan 𝜃 =
cot 𝜃
sin 𝜃 =
Domain Restrictions
𝜃 ≠ 𝑛𝜋
𝑛𝜋
2
𝑛𝜋
𝜃≠
2
𝜃≠
𝑛 = 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝑛 = 𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝑛 = 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
Quotient (or Ratio) Identities
Quotient Identities
sin 𝜃
tan 𝜃 =
cos 𝜃
Domain Restrictions
cos 𝜃 ≠ 0
𝑜𝑟
𝜃≠
cos 𝜃
cot 𝜃 =
sin 𝜃
sin 𝜃 ≠ 0
𝑜𝑟
𝜃 ≠ 𝑛𝜋
𝑛𝜋
2
𝑛 = 𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝑛 = 𝑖𝑛𝑡𝑒𝑟𝑔𝑒𝑟
Pythagorean Identities
𝒔𝒊𝒏𝟐 𝜽 + 𝒄𝒐𝒔𝟐 𝜽 = 𝟏
𝒕𝒂𝒏𝟐 𝜽 + 𝟏 = 𝒔𝒆𝒄𝟐 𝜽
𝟏 + 𝒄𝒐𝒕𝟐 𝜽 = 𝒄𝒔𝒄𝟐 𝜽
Negative Arguments Identities
𝒔𝒊𝒏 −𝜽 = −𝒔𝒊𝒏 𝜽
𝒄𝒐𝒔 −𝜽 = 𝒄𝒐𝒔 𝜽
𝒕𝒂𝒏 −𝜽 = −𝒕𝒂𝒏 𝜽
Notes:
The real number x or θ in these
identities may be changed by other
angles such as α, β, γ, A, B, C,….
 The resulting identities may then be
called trigonometric identities.

Example:

Find the remaining circular functions of θ
using the fundamental identities, given
5
sin θ = and P(θ) ϵ II.
13
Simplifying Expressions
Examples:
Simplify the following expressions using the
fundamental identities.
1. tan3 x csc3 x
2. sec x • cos x – cos2 x
3. (csc2 x – 1)(sec2 x sin2 x)
𝑐𝑠𝑐 2 θ−𝑐𝑜𝑡 2 θ
4.
𝑡𝑎𝑛2 θ−𝑠𝑒𝑐 2 θ
(cos θ – 1)(cos θ + 1)
6. sin2 θ + cot2 θ sin2 θ
5.
Proving Identities
There is no exact procedure to be followed in
proving identities. However, it may be helpful to
express all the given functions in terms of sines
and cosines and then simplify.
 To establish an identity, we may use one of the
following:
1. Transform the left member into the exact form
of the right.
2. Transform the right into the exact form of the
left, or
3. Transform each side separately into the same
form.

Examples
1.
2.
1+cos θ
sin θ
cos θ+1
sin θ cos θ
Prove that
+ tan θ =
is
an identity.
Verify if tan2 β – sin2 β = tan2 β sin2 β is
an identity.
Exercises
−5
6
If sin θ =
and P(θ) is in quadrant IV, find
the other trigonometric function values of θ
using the fundamental identities.
2. Express cos θ (tan θ – sec θ) in terms of
sine and cosine using the fundamental
identities and then simplify the expression.
1.
3.
Show that
identity.
tan2
θ (1 +
cot2
1
θ)=
is an
2
1−sin θ
Exercises
4.
Simplify the following expressions.
a)
b)
sin x sec x
tan x
csc θ cot θ
tan θ sec θ
c) (sec x + tan x)(sec x – tan x)
𝑐𝑜𝑠 2 𝑥
d) 2 –
1−sin 𝑥
Do Worksheet 6
Sum and Difference Identities
𝒔𝒊𝒏 𝑨 + 𝑩 = 𝒔𝒊𝒏 𝑨 𝒄𝒐𝒔 𝑩 + 𝒄𝒐𝒔 𝑨 𝒔𝒊𝒏 𝑩
𝒔𝒊𝒏 𝑨 − 𝑩 = 𝒔𝒊𝒏 𝑨 𝒄𝒐𝒔 𝑩 − 𝒄𝒐𝒔 𝑨 𝒔𝒊𝒏 𝑩
𝒄𝒐𝒔 𝑨 + 𝑩 = 𝒄𝒐𝒔 𝑨 𝒄𝒐𝒔 𝑩 − 𝒔𝒊𝒏 𝑨 𝒔𝒊𝒏 𝑩
𝒄𝒐𝒔 𝑨 − 𝑩 = 𝒄𝒐𝒔 𝑨 𝒄𝒐𝒔 𝑩 + 𝒔𝒊𝒏 𝑨 𝒔𝒊𝒏 𝑩
𝒕𝒂𝒏 𝑨 + 𝑩 =
𝒕𝒂𝒏 𝑨 + 𝒕𝒂𝒏 𝑩
𝟏 − 𝒕𝒂𝒏 𝑨 𝒕𝒂𝒏 𝑩
𝒕𝒂𝒏 𝑨 − 𝑩 =
𝒕𝒂𝒏 𝑨 − 𝒕𝒂𝒏 𝑩
𝟏 + 𝒕𝒂𝒏 𝑨 𝒕𝒂𝒏 𝑩
Double-Angle Identities
Sine
𝒔𝒊𝒏 𝟐𝑨 = 𝟐𝒔𝒊𝒏 𝑨 𝒄𝒐𝒔 𝑨
Cosine
𝒄𝒐𝒔 𝟐𝑨 = 𝒄𝒐𝒔𝟐 𝑨 − 𝒔𝒊𝒏𝟐 𝑨
𝒄𝒐𝒔 𝟐𝑨 = 𝟏 − 𝟐𝒔𝒊𝒏𝟐 𝑨
𝒄𝒐𝒔 𝟐𝑨 = 𝟐𝒄𝒐𝒔𝟐 𝑨 − 𝟏
Tangent
𝟐 𝒕𝒂𝒏 𝑨
𝒕𝒂𝒏 𝟐𝑨 =
𝟏 − 𝒕𝒂𝒏𝟐 𝑨
Half-Angle Identities
Sine
𝑨
𝟏 − 𝒄𝒐𝒔 𝑨
𝒔𝒊𝒏
=±
𝟐
𝟐
Cosine
𝑨
𝟏 + 𝒄𝒐𝒔 𝑨
𝒄𝒐𝒔
=±
𝟐
𝟐
Tangent
𝑨
𝟏 − 𝒄𝒐𝒔 𝑨
𝒕𝒂𝒏
=±
𝟐
𝟏 + 𝒄𝒐𝒔 𝑨
𝒕𝒂𝒏
𝑨
𝒔𝒊𝒏 𝑨
=
𝟐
𝟏 + 𝒄𝒐𝒔 𝑨
𝒕𝒂𝒏
𝑨
𝟏 − 𝒄𝒐𝒔 𝑨
=
𝟐
𝒔𝒊𝒏 𝑨
Product-to-Sum and Sum-to-Product
Identities
Product-to-Sum Identities
𝟏
𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 = 𝒄𝒐𝒔 𝑨 + 𝑩 + 𝒄𝒐𝒔 𝑨 − 𝑩
𝟐
𝒔𝒊𝒏 𝑨 𝒔𝒊𝒏𝑩 =
𝟏
𝒄𝒐𝒔 𝑨 − 𝑩 − 𝒄𝒐𝒔 𝑨 + 𝑩
𝟐
𝒔𝒊𝒏𝑨 𝐜𝐨𝐬 𝑩 =
𝟏
𝒔𝒊𝒏 𝑨 + 𝑩 + 𝒔𝒊𝒏 𝑨 − 𝑩
𝟐
𝟏
𝒄𝒐𝒔𝑨 𝒔𝒊𝒏 𝑩 =
𝒔𝒊𝒏 𝑨 + 𝑩 − 𝒔𝒊𝒏 𝑨 − 𝑩
𝟐
Product-to-Sum and Sum-to-Product
Identities
Sum-to-Product Identities
𝒔𝒊𝒏𝑨 + 𝒔𝒊𝒏𝑩 = 𝟐 𝒔𝒊𝒏
𝑨+𝑩
𝑨−𝑩
𝒄𝒐𝒔
𝟐
𝟐
𝒔𝒊𝒏𝑨 − 𝒔𝒊𝒏𝑩 = 𝟐 𝒔𝒊𝒏
𝑨−𝑩
𝑨+𝑩
𝒄𝒐𝒔
𝟐
𝟐
𝒄𝒐𝒔𝑨 + 𝒄𝒐𝒔𝑩 = 𝟐𝒄𝒐𝒔
𝑨+𝑩
𝑨−𝑩
𝒄𝒐𝒔
𝟐
𝟐
𝒄𝒐𝒔𝑨 − 𝒄𝒐𝒔𝑩 = −𝟐 𝒔𝒊𝒏
𝑨+𝑩
𝑨−𝑩
𝒔𝒊𝒏
𝟐
𝟐
Examples
1.
2.
3.
4.
5.
6.
Find the exact value of sin 75° using sum
and difference identities.
Simplify sin 20°cos 40° + cos 20°sin 40°.
Simplify tan(x + 4π).
−5
6
Given that cot θ =
and θ is in the
second quadrant, find:
a) sin 2θ b) tan 2θ c) cos 2θ
Find the exact value of sin 22.5° using
half-angle identities.
Simplify cot (90° - θ) if tan θ =
cofunction identities.
21
5
using
7.
8.
9.
Evaluate tan 165°.
Find the exact value of 105°.
Simplify cot (
π
3
- x ).
10. Simplify
the following trigonometric
expressions:
a)
b)
cos 3𝑥 −cos 𝑥
sin 3𝑥 +sin 𝑥
cos 5𝑥 +cos 2𝑥
sin 5𝑥 −sin 2𝑥