Trigonometric Identities. Uses and examples.

1. Trigonometric
Identities

2. Trigonometric Identity
Equalities that involve trigonometric
functions and are true for every single
value of the occurring variables.
 Identities involving certain functions of
one or more angles.

3. 3 Groups or Relation
Reciprocal Relation
Quotient Relation
Pythagorean Relation

4. Reciprocal Relation
The inverse trigonometric
functions are partial inverse
functions for the
trigonometric functions.

5. tan𝜃 =
y
x
and cot𝜃 =
x
y
therefore, tanθ and cotθ
are reciprocals of each other. The same thing
can be said about sinθ and cscθ as well as cosθ
and secθ.
𝒄𝒐𝒕𝜽 =
𝟏
𝒕𝒂𝒏𝜽
𝒔𝒆𝒄𝜽 =
𝟏
𝒄𝒐𝒔𝜽
𝒄𝒔𝒄𝜽 =
𝟏
𝒔𝒊𝒏𝜽

6. Since the product of a number and its
reciprocal equals 1, these relations may
also be written as:
tanθcotθ=1
cosθsecθ=1
sinθcscθ=1

7. Quotient
Relation

8. Simplifying,
sinθ
cosθ
=
y
x
. But 𝑡𝑎𝑛𝜃 =
y
x
.
So by transivity;
𝒕𝒂𝒏𝜽 =
𝒔𝒊𝒏𝜽
𝒄𝒐𝒔𝜽

9. Since cotθ is the reciprocal of tanθ the
quotient can be derived to get
𝒄𝒐𝒕𝜽 =
𝒄𝒐𝒔𝜽
𝒔𝒊𝒏𝜽

10. Pythagorean
Relation
 The basic relationship between the sine and
the cosine is the Pythagorean trigonometric
identity:
where cos2 θ means (cos(θ))2 and sin2 θ
means (sin(θ))2.
 This can be viewed as a version of
the Pythagorean theorem, and follows from
the equation x2 + y2 = 1for the unit circle.

11. By Pythagorean Theorem, 𝑥2
+ 𝑦2
=
𝑟2
. Dividing both members by r²
results to
x2
r2 +
y2
𝑟2 = 1. Since 𝑐𝑜𝑠𝜃 =
x
r
and 𝑠𝑖𝑛𝜃 =
y
r
, then,
cos²θ + sin²θ=1

12. Dividing both members or 𝒙 𝟐
+ 𝒚 𝟐
= 𝒓 𝟐
by x²
you get;
1 + tan²θ = sec²θ

13. dividing by y², you get;
cot²θ + 1 = csc²θ

14. Activity

15. A. Fill in the blanks to complete the table.
The Fundamental Trigonometric Identities and Their Alternate Forms
sinθcscθ = 1 1.
𝑠𝑖𝑛𝜃 =
1
cscθ
2.
𝑠𝑖𝑛𝜃 =
1
cosθ
𝑐𝑜𝑠𝜃 =
1
secθ
tanθcotθ = 1
𝑐𝑜𝑡𝜃 =
1
tanθ
3.
4.
𝑐𝑜𝑡𝜃 =
1
tanθ
5.
𝑐𝑜𝑡𝜃 =
sinθ
cosθ
6. 7.
sin²θ + cos²θ = 1 8. cos²θ = 1 - sin²θ
9. tan²θ = sec²θ - 1 sec²θ - tan²θ = 1
1 + cot²θ = csc²θ cot²θ = csc²θ - 1 10.

16. B. Use the fundamental identities to find
the values of the other trigonometric
functions.
1. tanθcotθ = ___________
2. csc²θ = ____________
3.
sinθ
cosθ
= ___________
4. cosθ = ____________
5. sinθ = ___________

17. Assignment
What are the terminologies
used in the graphs of
trigonometric function?
Define each.
Reference: Trigonometry
pages 141-142