TRIGONOMETRIC IDENTITIES

TRIGONOMETRIC IDENTITIES

DEFINITION: A trigonometric identity is an equation involving the trigonometric functions that holds for all values of the variable.

BASIC TRIGONOMETRIC IDENTITIESReciprocal Identities

cos1sec

tan1cot

sin1 csc 1cossin 22

22 sec1tan

22 csc1cot

Pythagorean Identities

sec1cos

csc1sin

cot1tan

cossin tan

sincos cot

sin sin

Negative Argument Identities

cos cos

tan tan

Quotient or Ratio Identities

There is no set procedure to prove identities. However, there are several strategies to use when proving identities.

1. Know the fundamental identities and look for ways to apply them.

2. Write all expressions in terms of sine and cosine.3. If you choose to work with only one side of an

identity, continuously refer back to the other side to see what you are trying to obtain.

4. When one side contains only one trigonometric function, attempt to rewrite all the functions on the other side in terms of that function. It is usually easier to start with the more complicated side.

5. Use Pythagorean identities to substitute for the expression equal to 1.

6. Perform algebraic operations.a) Factoring.b) Simplifying complex rational expressions. c) Finding the LCD and combining fractions.d) Combining like terms. e) Multiplying both the numerator and denominator by

the same expression to obtain an equivalent fraction. f) Replacing a binomial with a monomial.

Note: Proving an identity is not the same as solving an equation. This means you can’t perform operations such as adding the same expression to both sides or dividing both sides by the same expression. These operations apply only to an equation where the statement is known to be true; an identity must be proven to be true.

EXAMPLE:Prove the following identities.

sintan cos .a sincsccos cot .b

cos1

sin-1cos .c 2

tansin

csccottansin .d

cos2

sin1cos

sin1cos .e

322 sincos

tansinsecsin .i

222 1costantansin .g

csc2

sincos1

cos1sin .j

cossin1

sin-1cos .k

1sectan

coscos1 .l

2

2244 cossincossin .f

22 sin2tantancossin .h

Sum and Difference Identities

BtanAtan1

BtanAtanBA tan

BsinAcosBcosAsinBA sin

BsinAcosBcosAsinBA sin BsinAsinBcosAcosBA cos

BsinAsinBcosAcosBA cos

BtanAtan1

BtanAtanBA tan

EXAMPLE:

0105 cos a)

I. Use sum or difference identities to find the exact value of the given function.

015 sinb) 0375 tan c)

II. Given that , , , and 5

3A sin 5

12B tan 2

A0

23B , find the following:

B-A tan c) B-A cos b) BA sina)

.lies B-A of sideterminal the whichin quadrant the d)

0000 25sin80cos25cos80 sina)

III. Write each expression in terms of a trigonometric function of one angle.

sinsin2cos-cos a)

3 sin sin 3 cos cos b)

4tan

43tan-1

4tan

43tan

c)

IV. Prove each identity.

cos270 sinb) 0

tan360 tan c) 0

Double - Angle Identities

Atan1A tan 22A tan 2

Acos Asin 22A sin Asin Acos2A cos 22

1Acos 22A cos 2 Asin 212A cos 2

I. Write each expression in terms of a trigonometric function of one angle.

EXAMPLE:

00 35cos35 sin2 a)02 402sin-1 b)

02

0

5.22tan15.22 tan 2 c)

2cos find ,900 and 53 sinIf a) 00

II. Use double-angle identities to find the exact value of the given function.

2

2

sectan-12 cos a)

2sin find ,18009 and 34 tan If b) 00

2tan find ,360027 and 135cos If c) 00

III. Prove each identity.

cossinsin2coscos2sin

cos1 b)

44 sincos2 cos c)

tancottancot2 secd)

Half - Angle Identities

2Acos1

2Asin

I. Use half-angle identities to find the exact values of the following.

EXAMPLE:

2Acos1

2Acos

1Acos , Acos1Acos1

2Atan

0Asin , Asin

Acos12Atan

1Acos , Acos1

Asin2Atan

0105 tan )a 022.5 sin)b

87 cos )c

12 sin)d

.II quadrant in lies and 2tan if 2

tan a)

.III quadrant in lies and 1312 sinif

2cos b)

.III quadrant in lies and 1312cos2 if

2cos c)

II. Find the exact value of each trigonometric function. Assume .00 3600

Product / Sum Identities BA cos BA cosBcos Acos 2

BA cos BA cosB sin A sin 2 BA sin BA sinBcos A sin 2 BA sin BA sinBsin A cos 2

2ZW sin

2ZW cos 2 Z sin W sin

2ZW cos

2ZW sin2 Z sin W sin

2ZW sin

2ZW sin 2Z cos W cos

2ZW cos

2ZW cos 2Z cos W cos

B AZ and B AW Let

EXAMPLE:I. Express each product as a sum or difference.

00 20 cos50 sin2 )a 00 10 cos40 cos 2 )a

II. Express each sum or difference as a product.00 10cos 70 cos )a

8x cos - 4x cos )b

4x cos 12x cos )c

6x sin- 10x sin )b

III. Express each sum or difference as a product.

8 sin-2 sin2 cos-8 cos5 tan )a

3 sin-5 sin3 cos5 cos cot )b