The document provides an overview of trigonometry, focusing on the relationships between the sides and angles of right-angled triangles, and introduces key concepts such as trigonometric ratios (sine, cosine, tangent) and identities. It details methods for proving trigonometric identities, the application of exact values, and their use in various mathematical fields. Additionally, it highlights the broad applications of trigonometry across numerous scientific disciplines.

1. Trigonometry
Class:IX
1

2.  Trigonometry is the
branch of mathematics
which deals with
triangles, particularly
triangles in a plane
where one angle of the
triangle is 90 degrees.
.
Trigonometry
2

3.  Trigonometry specifically deals with
 the relationships between the
 sides and the angles of triangles.
----------------------------------------
--------
3

4. In trigonometry, the ratio we are
talking about is the comparison of
the sides of a RIGHT ANGLED
TRIANGLE. Two things MUST BE understood:
1. This is the hypotenuse.
.
2. This is 90°

5. Now that we agree about the hypotenuse
and right angle, there are only 4 things left;
the 2 other angles and the 2 other sides.
A
.
Opposite side
Adjacent side
Hypotenuse

6. Remember we use the
right angle
√

7. One more thing…
θ this is the symbol for an unknown angle
measure.
It’s name is ‘Theta’.

8. Trigonometric Ratios
Name
“say” Sine Cosine tangent
Abbreviation
Sin Cos Tan
Ratio of an
angle
measure
Sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side

9. One more
time…
Here are the
ratios:
sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side

10. Trigonometric Identities
A trigonometric equation is an equation that involves
at least one trigonometric function of a variable. The
equation is a trigonometric identity if it is true for all
values of the variable for which both sides of the
equation are defined.
Prove that tan 
sin
cos
.

y
x

y
r

x
r

y
r

r
x

y
x
L.S. = R.S.
5.4.2
Recall the basic
trig identities:
sin 
y
r
cos 
x
r
tan 
y
x

11. 5.4.3
Trigonometric Identities
Quotient Identities
tan 
sin
cos
cot 
cos
sin
Reciprocal Identities
sin 
1
csc
cos 
1
se c
tan 
1
cot
Pythagorean Identities
sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2
sin2 = 1 - cos2
cos2 = 1 - sin2
tan2 = sec2 - 1 cot2 = csc2 - 1

12. Trigonometric Identities [cont’d]
sinx x sinx = sin2x
cos 
1
cos

cos2
cos

1
cos

cos2  1
cos
sinA cos A2
 sin2 A 2sinAcos A cos2 A
 12sinAcosA
cosA
sinA
1
sinA

cosA
sinA

sinA
1
= cosA
5.4.4

13. Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify.
c os sin tan
 cos  sin
sin
cos
 cos 
sin2 
cos

cos 2  sin2
cos

1
cos
 sec
a)
b)
cot2 
1  sin2 

cos 2
sin2 
cos 2
1

1
sin2 
 csc2 
5.4.5

cos2
sin2 

1
cos2 

14. 5.4.6
Simplifing Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x
 1  2 tanx  tan2 x  2
sinx
cosx
 1 tan2 x  2tanx  2tanx
 sec2 x
d)
cscx
tanx  cotx

1
sinx
sinx
cosx

cosx
sinx

1
sinx
sin2 x  cos 2 x
sinxcos x


1
sinx
1
sinx cosx
1
sinx

sinx cos x
1
 cos x
 (1  tanx)2  2 sinx
1
cosx

15. 5.4.7
Proving an Identity
Steps in Proving Identities:
1. Start with the more complex side of the identity and work
with it exclusively to transform the expression into the
simpler side of the identity.
2. Look for algebraic simplifications:
• Do any multiplying , factoring, or squaring which is
obvious in the expression.
• Reduce two terms to one, either add two terms or
factor so that you may reduce.

16. 16
3. Look for trigonometric simplifications
• Look for familiar trig relationships :
• If the expression contains squared terms
• , think of the Pythagorean Identities.
Transform each term to sine or cosine, if the
expression cannot be simplified easily using
other ratios.

17. 5.4.8
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x
= sec x + 1
1 + sec x
L.S. = R.S.
b) sec x = tan x csc x

sinx
cos x

1
sinx

1
cosx
 secx
secx
L.S. = R.S.
c) tan x sin x + cos x = sec x

sinx
cosx

sinx
1
 cosx

sin2 x  cos 2 x
cos x

1
cosx
 secx
secx
L.S. = R.S.

18. d) sin4x - cos4x = 1 - 2cos2 x
= (sin2x - cos2x)(sin2x + cos2x)
= (1 - cos2x - cos2x)
= 1 - 2cos2x
1 - 2cos2x
L.S. = R.S.
e)
1
1  cosx

1
1  cosx
 2 csc2 x

(1  cosx)  (1  cosx)
(1  cosx)(1  cosx)

2
(1  cos2 x)

2
sin2 x
 2csc2 x
2csc2 x
L.S. = R.S.
Proving an Identity
5.4.9

19. Proving an Identity
5.4.10
f)
cosA
1  sinA

1  sinA
cos A
 2 secA

cos2 A  (1  sinA)(1  sinA)
(1  sinA)(cosA)

cos2 A  (1  2sinA sin2 A)
(1  sinA)(cosA)

cos2 A  sin2 A1  2sinA
(1  sinA)(cosA)

2  2sinA
(1  sinA)(cosA)

2(1  sinA)
(1  sinA)(cosA)

2
(cosA)
 2secA
2secA
L.S. = R.S.

20. Using Exact Values to Prove an Identity
5.4.11
Consider sinx
1  cos x

1  cosx
sinx
.
a) Use a graph to verify that the equation is an identity.
b) Verify that this statement is true for x =

6
.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
y 
1  sinx
cosx
1  sinx
cosx
a)

21. Using Exact Values to Prove an Identity [cont’d]
b) Verify that this statement is true for x =
sinx
1  cosx

1  cosx
sinx

1
2
1 
3
2

6
.

sin

6
1  cos

6

1
2

2
2  3

1
2  3

1  cos

6
sin

6
3
2
1
2
 1 

2  3
2

2
1
 2  3
 2  3
Rationalize the
denominator:

1
2  3
1
2  3

2  3
2  3

2  3
4  3
 2  3
L.S. = R.S.
Therefore, the identity is
true for the particular
case of x 
5.4.12

6
.

22. Using Exact Values to Prove an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
5.4.13
sinx
1  cosx

1  cosx
sinx

sinx
1  cosx

1  cosx
1  cosx

sinx(1  cosx)
1  cos2 x

sinx(1  cosx)
sin2 x

1  cosx
sinx
1  cosx
sinx
L.S. = R.S.
Restrictions:
Note the left side of the
equation has the restriction
1 - cos x ≠ 0 or cos x ≠ 1.
Therefore, x ≠ 0 + 2 n,
where n is any integer.
The right side of the
equation has the restriction
sin x ≠ 0. x = 0 and 
Therefore, x ≠ 0 + 2n
and x ≠  + 2n, where
n is any integer.

23. Proving an Equation is an Identity
Consider the equation sin2 A1
sin2 A sinA
 1 
1
sinA
.
a) Use a graph to verify that the equation is an identity.
b) Verify that this statement is true for x = 2.4 rad.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
y 
sin2 A 1
sin2 A sinA
1 
1
sinA
a)
5.4.14

24. Proving an Equation is an Identity [cont’d]
b) Verify that this statement is true for x = 2.4 rad.
sin2 A1
sin2 A sinA
 1 
1
sinA

(s in 2.4)2  1
(s in 2.4)2  sin2.4
= 2.480 466
 1 
1
sin 2.4
= 2.480 466
L.S. = R.S.
Therefore, the equation is true for x = 2.4 rad.
5.4.15

25. 5.4.16
Proving an Equation is an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
sin2 A1
sin2 A sinA
 1 
1
sinA

(s inA1)(sinA 1)
sinA(s inA1)

(sinA1)
sinA

sinA
sinA

1
sinA
 1 
1
sinA
1 
1
sinA
L.S. = R.S.
Note the left side of the
equation has the restriction:
sin2A - sin A ≠ 0
sin A(sin A - 1) ≠ 0
sin A ≠ 0 or sin A ≠ 1
A  0, or A

2
Therefore A,  0  2 n or
A  + 2n, or
A 

2
 2 n, wheren is
any integer.
The right side of the
equation has the restriction
sin A ≠ 0, or A ≠ 0.
Therefore, A ≠ 0,  + 2 n,
where n is any integer.

26. Applications of Trigonometry
 This field of mathematics can be applied in
astronomy,navigation, music theory, acoustics, optics,
analysis of financial markets, electronics, probability
theory, statistics, biology, medical imaging (CAT scans
and ultrasound), pharmacy, chemistry, number theory
(and hence cryptology), seismology, meteorology,
oceanography and in many physical sciences.
26

27. Trigonometry is a branch of Mathematics
with several important and useful
applications. Hence it attracts more and
more research with several theories
published year after year
27
Conclusion