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
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