Trigonometry

Trigonometric Identities

Trigonometric Identities

An identity is an equation which is true for all values of the variable.

There are many trig identities that are useful in changing the appearance of an expression.

The most important ones should be committed to memory.

Trigonometric Identities

Reciprocal Identities

1sin

cosecx

x

1cos

secx

x

1tan

cotx

x

sintan

cos

xx

x

coscot

sin

xx

x

Quotient Identities

cos2θ + sin2θ = 1

θ

(x, y)

r y

x

By Pythagoras’ Theorem

x 2 + y

2 = r 2

Divide both sides by r2 2

2 21

x y

r r

cosx

r sin

y

r

2 2cos sin 1

Trigonometric Identities

Pythagorean Identities

The fundamental Pythagorean identity2 2sin cos 1x x

2 21 cot cosecx x Divide by sin2 x

Divide by cos2 x 2 2tan 1 secx x

Identities involving Cosine Rule Using the usual notation for a triangle, prove that c(bcosA – acosB) = b2 – a2

2 2 2 2 cosb a c ac B 2 2 2

cos2

b c aA

bc

2 2 2

cos2

a c bB

ac

( cos cos ) cos cosc b A a B bc A ac B

2 2 2 2 2 2

2 2

b c a a c bbc ac

bc ac

2 2 2 2 cosa b c bc A

2 2 2 2 2 2

2

b c a a c b

( cos cos )c b A a B2 2 2 2 2 2

2 2

b c a a c b

2 22 2

2

b a 2 2b a

Using the usual notation for a triangle, prove that c(bcosA – acosB) = b2 – a2

Identities involving Cosine Rule

Trigonometric Formulas

cos( ) cos

sin( ) sin

tan( ) tan

A A

A A

A A

Page 9 of tables

cos( ) cos cos sin sinA B A B A B

Trigonometric Formulas

2

2

cos sin

sin cos

A A

A A

cos( ) cos cos sin sinA B A B A B

cos ( ) cos cos( ) sin sin( )A B A B A B

Replace B with – B

cos cos cos sin ( sin )A B A B A B

cos cos cos sin sinA B A B A B

Prove cos cos cos sin sinA B A B A B

cos( ) cos cos sin sinA B A B A B

cos( ) cos cos sin sinA A A A A A

Replace B with A

2 2cos2 cos sinA A A

2 2Prove cos2 cos sinA A A

cos( ) cos cos sin sinA B A B A B

2 2 2cos ( ) cos( )cos sin( )sinA B A B A B

2cos ( ( ) sin cos cos sinA B A B A B

Prove sin( ) sin cos cos sinA B A B A B

Replace A with – A2

sin( ) sin cos cos sinA B A B A B

Solving Trig Equations To solve trigonometric equations:

If there is more than one trigonometric function, use identities to simplify

Let a variable represent the remaining function

Solve the equation for this new variable

Reinsert the trigonometric function

Determine the argument which will produce the desired value

cos2A = (1 + cos 2A)12

(1 – cos2A)

(i) Using cos 2A = cos2A – sin2A, or otherwise, prove12

cos2A = (1 + cos 2A).

2005 Paper 2 Q4 (b)

cos 2A = cos2A – sin2A

cos 2A = cos2A – 1 + cos2A

cos 2A = 2cos2A – 11 +

= cos x

(ii) Hence, or otherwise, solve the equation

1 + cos 2x = cos x, where 0º ≤ x ≤ 360º.

2005 Paper 2 Q4 (b)

1 + cos 2x

2cos2x – cos x = 0

2cos2x

cos x(2cos x – 1) = 0

From (i)

360º180º

1

–1cos x = 0

90 ,270x

1cos

2x

60 ,300x

23 3sin 2cos 0x x

2Solve 3 3sin 2cos 0, where 0 2 .x x x

23 3sin 2 1 sin 0x x

23 3sin 2 2sin 0x x

21 3sin 2sin 0x x

22sin 3sin 1 0x x

2 2cos 1 sinx x

Expand

Collect like terms

Rearrange

Let sint x

22 3 1 0t t Factorise

(2 1)( 1) 0t t

2 1 0 1 0t t

2 1 t Replace t with sin x

1sin

2x

1

2t

1 t

sin 1x

5 or

6 6x

2x

2Solve 3 3sin 2cos 0, where 0 2 .x x x

2ππ

1

–1