Trignometry

TrigonometryClass:IX

1

Trigonometry

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

.

2

------------------------------------------------

Trigonometry specifically deals with

the relationships between the

sides and the angles of triangles.

3

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°

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

Ad

jace

nt

sid

e

Hypotenuse

Remember we use the right angle

√

θ this is the symbol for an unknown angle

measure.

It’s name is ‘Theta’.

One more thing…

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

One more

time…

Here are the

ratios:

sinθ = opposite side

hypotenuse

cosθ = adjacent side

hypotenuse

tanθ =opposite side

adjacent side

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.

Trigonometric Identities

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

5.4.3

Trigonometric Identities

Quotient Identities

tan sin

coscot

cos

sin

Reciprocal Identities

sin 1

csccos

1

sectan

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

sinx x sinx = sin2x

cos 1

cos

cos2

cos

1

cos

cos2 1

cos

sinA cos A 2

sin2

A 2sinAcos A cos2

A

12sinAcosA

cosA

sinA1

sinA

cosA

sinA

sinA

1

= cosA

Trigonometric Identities [cont’d]

5.4.4

Identities can be used to simplify trigonometric expressions.

Simplifying Trigonometric Expressions

cos sin tan

cos sin

sin

cos

cos

sin2

cos

cos 2 sin2

cos

1

cos

sec

a)

Simplify.

b)cot2

1 sin2

cos 2

sin2 cos

2

1

1

sin2

csc2

5.4.5

cos2

sin2

1

cos2

5.4.6

Simplifing Trigonometric Expressions

c) (1 + tan x)2 - 2 sin x sec x

1 2 tanx tan2x 2

sinx

cosx

1 tan2x 2tanx 2tanx

sec2x

d)cscx

tanx cotx

1

sinx

sinx

cosx

cosx

sinx

1

sinx

sin2x cos

2x

sinxcos x

1

sinx

sinx cos x

1

cos x

1

sinx

1

sinx cosx

(1 tanx)2

2 sinx1

cosx

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

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.

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.

d) sin4x - cos4x = 1 - 2cos2 x

= (sin2x - cos2x)(sin2x + cos2x)

= (1 - cos2x - cos2x)

= 1 - 2cos2x

L.S. = R.S.

1 - 2cos2x

e)

1

1 cosx

1

1 cosx 2 csc

2x

(1 cosx) (1 cosx)

(1 cosx)(1 cosx)

2

(1 cos2

x)

2

sin2x

2csc2x

2csc2x

L.S. = R.S.

Proving an Identity

5.4.9

Proving an Identity

5.4.10

f)

cos A

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.

Using Exact Values to Prove an Identity

5.4.11

Consider sinx

1 cos x

1 cosx

sinx.

b) Verify that this statement is true for x =

6.

a) Use a graph to verify that the equation is an identity.

c) Use an algebraic approach to prove that the identity is true

in general. State any restrictions.

y 1 cosx

sinxy

sinx

1 cosxa)

sinx

1 cosx

1 cosx

sinx

1

2

1 3

2

b) Verify that this statement is true for x =

6.

sin

6

1 cos

6

1

2

2

2 3

1

2 3

1 cos

6

sin

6

1 3

2

1

2

2 3

2

2

1

2 3

2 3

1

2 3

2 3

2 3

2 3

4 3

2 3

Rationalize the

denominator:

1

2 3

L.S. = R.S.

Using Exact Values to Prove an Identity [cont’d]

5.4.12

Therefore, the identity is

true for the particular

case of x

6.



Using Exact Values to Prove an Identity [cont’d]

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)

sin2x

1 cosx

sinx

1 cosx

sinx

L.S. = R.S.

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


sin x ≠ 0. x = 0 and Therefore, x ≠ 0 + 2n

and x ≠ + 2n, where

n is any integer.

Restrictions:

Proving an Equation is an Identity

Consider the equation sin2 A 1

sin2

A sinA 1

1

sinA.

b) Verify that this statement is true for x = 2.4 rad.

a) Use a graph to verify that the equation is an identity.



y sin2 A 1

sin2

A sinAy 1

1

sinA

a)

5.4.14

b) Verify that this statement is true for x = 2.4 rad.

Proving an Equation is an Identity [cont’d]

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

Therefore, the equation is true for x = 2.4 rad.

L.S. = R.S.

5.4.15

5.4.16

Proving an Equation is an Identity [cont’d]

sin2 A 1

sin2

A sinA 1

1

sinA



(s inA 1)(sinA 1)

sinA(s inA 1)

(sinA1)

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

A 0, or A

2

Therefore, A 0 2 n or

A + 2n, or

A

2 2 n, where n is

any integer.

The right side of the


sin A ≠ 0, or A ≠ 0.

Therefore, A ≠ 0, + 2 n,

where n is any integer.

sin A(sin A - 1) ≠ 0

sin A ≠ 0 or sin A ≠ 1

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

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

Trignometry

Education

Transcript of Trignometry