Chapter 11: Trigonometric Identities

11.1 Trigonometric Identities

11.2 Addition and Subtraction Formulas

11.3 Double-Angle, Half-Angle, and Product-Sum Formulas

11.4 Inverse Trigonometric Functions

11.5 Trigonometric Equations

11.1 Trigonometric Identities

• In 1831, Michael Faraday discovers a small electric current when a wire is passed by a magnet.

• This phenomenon is known as Faraday’s law. This property is used to produce electricity by rotating thousands of wires near large electromagnets.

• Electric generators supply electricity to most homes at a rate of 60 cycles per second.

• This rotation causes alternating current in wires and can be modeled by either sine or cosine functions.

11.1 Fundamental Identities

Fundamental Identities

Pythagorean Identities

Even-Odd Identities

Note It will be necessary to recognize alternative forms of the identities above, such as sin² = 1 – cos² and cos² = 1 – sin² .

sin1

csccos

1sec

tan1

cot

sincos

cotcossin

tan

222222 csccot1sec1tan1cossin

tan)tan(cos)cos(sin)sin(

11.1 Fundamental Identities

Cofunction Identities

uuuuuu csc2

seccot2

tancos2

sin

uuuuuu sec2

csc tan2

cotsin2

cos

11.1 Looking Ahead to Calculus

• Work with identities to simplify an expression with the appropriate trigonometric substitutions.

• For example, if x = 3 tan , then

29 x 2)tan3(9

2tan99 )tan1(9 2

2tan13

2sec3 20for sec3

11.1 Expressing One Function in Terms of AnotherExampleExpress cos x in terms of tan x.

SolutionSince sec x is related to both tan x and cos x by identities, start with tan² x + 1 = sec² x.

Choose + or – sign, depending on the quadrant of x.

xx 22 sec

1

1tan

1

xx

22

cos1tan

1

1tan

1cos

2

xx

1tan

1tancos

2

2

x

xx

xx 22 sec1tan

11.1 Rewriting an Expression in Terms of Sine and CosineExampleWrite tan + cot in terms of sin and cos .Solution

cottan

sin

cos

cos

sin

sincos

cos

sincos

sin 22

sincos

cossin 22

sincos

1

11.1 Verifying Identities1. Learn the fundamental identities.2. Try to rewrite the more complicated side of the equation so that

it is identical to the simpler side.3. It is often helpful to express all functions in terms of sine and

cosine and then simplify the result.4. Usually, any factoring or indicated algebraic operations should

be performed. For example,

5. As you select substitutions, keep in mind the side you are not changing, because it represents your goal.

6. If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin² x, which could be replaced with cos² x.

. and ,)1(sin1sin2sin cossinsincos

cos1

sin122

11.1 Verifying an Identity ( Working with One Side)ExampleVerify that the following equation is an identity.

cot x + 1 = csc x(cos x + sin x)

Solution Since the side on the right is more complicated, we work with it.

Original identity

Distributive property

The given equation is an identity because the left side equals the right side.

xx

sin

1csc

1sin

sin and

sin

coscot

x

x

x

xx

)sin(coscsc1cot xxxx )sin(coscsc xxxRHS

)sin(cossin

1xx

x

x

x

x

x

sin

sin

sin

cos

LHSx 1cot

11.1 Verifying an IdentityExample Verify that the following equation is an identity.

Solution

tttttt 22 cscsec

cossincottan

tttt

cossincottan

ttt

ttt

cossincot

cossintan

ttt

ttt

cossin1cot

cossin1tan

tttt

tttt

cossin1

sincos

cossin1

cossin

tt 22 sin

1

cos

1 tt 22 cscsec

11.1 Verifying an Identity ( Working with Both Sides)ExampleVerify that the following equation is an identity.

Solution

2

2

cos

sinsin21tansectansec

tansectansec

LHS

cos)tan(seccos)tan(sec

costancosseccostancossec

costan1costan1

coscossin

1

coscossin

1

sin1sin1

2

2

cossinsin21 RHS

sin1sin1

2

2

cos

)sin1(

2

2

sin1

)sin1(

)sin1)(sin1()sin1( 2

11.1 Verifying an Identity ( Working with Both Sides)

We have shown that

.cos

sinsin21sin1sin1

tansectansec

2

2

11.1 Applying a Pythagorean Identity to RadiosExample Tuners in radios select a radio station by adjusting the frequency. These tuners may contain an inductor L and a capacitor C. The energy stored in the inductor at time t is given by:

L(t) = k sin² (2Ft)

and the energy stored in the capacitor is given by

C(t) = k cos² (2Ft),

where F is the frequency of the radio station and k is a constant. The total energy E in the circuit is given by E(t) = L(t) + C(t). Show that E is a constant function.

11.1 Applying a Pythagorean Identity to Radios

Solution

k)1(k

))2(cos)2((sin 22 FtFtk )2(cos)2(sin 22 FtkFtk

)()()( tCtLtE