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TOPIC 3: TRIGONOMETRY II
PART A: FUNDAMENTAL TRIGONOMETRIC IDENTITIES
Memorize these in both “directions” (i.e., left-to-right and right-to-left).
Reciprocal Identities
csc x =
sin x
sec x =
cos x
cot x =
tan x
sin x =
csc x
cos x =
sec x
tan x =
cot x
WARNING 1: Remember that the reciprocal of sin x is csc x , not sec x.
TIP 1: We informally treat “0” and “undefined” as reciprocals when we are
dealing with basic trigonometric functions. Your algebra teacher will not
want to hear this, though!
Quotient Identities
tan x =
sin x
cos x
and cot x =
cos x
sin x
Pythagorean Identities
sin 2 x + cos 2 x = 1
1 + cot 2 x = csc 2 x
tan 2 x + 1 = sec 2 x
TIP 2: The second and third Pythagorean Identities can be obtained from the
first by dividing both of its sides by sin 2 x and cos 2 x , respectively.
TIP 3: The squares of csc x and sec x , which have “Up-U, Down-U”
graphs, are all alone on the right sides of the last two identities. They can
never be 0 in value. (Why is that? Look at the left sides.)
Cofunction Identities
If x is measured in radians, then:
sin x = cos
x
cos x = sin
x
We have analogous relationships for tangent and cotangent, and for
secant and cosecant; remember that they are sometimes undefined.
Think: Cofunctions of complementary angles are equal.
Even / Odd (or Negative Angle) Identities
Among the six basic trigonometric functions, only cosine (and its
reciprocal, secant) are even :
cos (^) ( x ) = cos x sec (^) ( x ) = sec x
However, the other four are odd :
sin (^) ( x ) = sin x csc (^) ( x ) = csc x
tan (^) ( x ) = tan x cot (^) ( x ) = cot x
- If f is an even function, then the graph of y = f (^) ( x ) is symmetric about the
y -axis.
- If f is an odd function, then the graph of y = f (^) ( x ) is symmetric about the
origin.
- Domain for tangent: The “ X ”s on the unit circle below correspond to an
undefined slope. Therefore, the corresponding real numbers (the corresponding
angle measures in radians) are excluded from the domain.
- Domain for tangent and secant: The “ X ”s on the unit circle above also correspond
to a cosine value of 0. By the Quotient Identity for tangent tan =
sin
cos
^
and the
Reciprocal Identity for secant sec =
cos
^
, we exclude the corresponding
radian measures from the domains of both functions.
- Domain for cotangent and cosecant: The “ X ”s on the unit circle below
correspond to a sine value of 0. By the Quotient Identity for cotangent
cot =
cos
sin
^
and the Reciprocal Identity for cosecant csc =
sin
^
, we
exclude the corresponding radian measures from the domains of both functions.
- Range for cosecant and secant: We turn “inside out” the range for both sine and
cosine, which is 1, 1.
- Range for cotangent: This is explained by the fact that the range for tangent is
( ,^ ) and the Reciprocal Identity for cotangent:^ cot^ ^ =^
tan
^
. cot is 0 in
value tan is undefined.
PART C: GRAPHS OF THE SIX BASIC TRIGONOMETRIC FUNCTIONS
- The six basic trigonometric functions are periodic, so their graphs can be
decomposed into cycles that repeat like wallpaper patterns. The period for tangent
and cotangent is ; it is 2 for the others.
- A vertical asymptote (“VA”) is a vertical line that a graph approaches in an
“explosive” sense. (This idea will be made more precise in Section 2.4.) VAs on
the graph of a basic trigonometric function correspond to exclusions from the
domain. They are graphed as dashed lines.
- Remember that the domain of a function f corresponds to the x -coordinates picked up by the graph of y = f (^) ( x ) , and the range corresponds to the
y -coordinates.
- Remember that cosine and secant are the only even functions among the six, so
their graphs are symmetric about the y -axis. The other four are odd , so their
graphs are symmetric about the origin.
PART D: SOLVING TRIGONOMETRIC EQUATIONS
Example 1 (Solving a Trigonometric Equation)
Solve: 2sin 4( x ) = 3
§ Solution
2sin 4( x ) = 3 Isolate the sine expression.
sin 4( x ) =
Substitution: Let = 4 x.
sin =
We will now solve this equation for .
Observe that sin
, so
will be the reference angle for our solutions
for . Since
is a negative sine value, we want “coreference angles”
of
in Quadrants III and IV.
Our solutions for are:
+ 2 n , or =
From this point on, it is a matter of algebra.
To find our solutions for x , replace with 4 x , and solve for x.
4 x =
+ 2 n , or 4 x =
x =
^
n , or x =
^
n (^) ( n )
x =
n , or x =
n (^) ( n )
Solution set: x x =
n , or x =
n (^) ( n )
PART E: ADVANCED TRIGONOMETRIC IDENTITIES
These identities may be derived according to the flowchart below.
for cosine
GROUP 3a: DOUBLE-ANGLE (Think: Angle-Reducing, if u > 0) IDENTITIES
Memorize: (Also be prepared to recognize and know these “right-to-left.”)
sin 2( u ) = 2 sin u cos u
Think: “Twice the product” Reading “right-to-left,” we have:
2 sin u cos u = sin 2( u )
(This is helpful when simplifying.)
cos 2( u ) = cos 2 u sin 2 u
Think: “Cosines – Sines” (again) Reading “right-to-left,” we have:
cos 2 u sin 2 u = cos 2( u )
Contrast this with the Pythagorean Identity:
cos 2 u + sin^2 u = 1
tan 2( u ) = 1 2 tan tan^2 u u
(Hard to memorize; we’ll show how to obtain it.)
Notice that these identities are “angle-reducing” (if u > 0) in that they allow you to go from trigonometric functions of (2 u ) to trigonometric functions of simply u.
Obtaining the Double-Angle Identities from the Sum Identities:
Take the Sum Identities, replace v with u , and simplify.
sin 2( u ) = sin ( u + u )
= sin u cos u + cos u sin u (From Sum Identity) = sin u cos u + sin u cos u (Like terms!!) = 2 sin u cos u
cos 2( u ) = cos ( u + u )
= cos u cos u sin u sin u (From Sum Identity) = cos 2 u sin 2 u
tan 2( u ) = tan ( u + u )
= 1 tan tan^ u^ + u^ tan tan^ u u (From Sum Identity)
= 2 tan^ u 1 tan 2 u
This is a “last resort” if you forget the Double-Angle Identities, but you will need to recall the Double-Angle Identities quickly!
One possible exception: Since the tan 2( u ) identity is harder to remember, you may prefer
to remember the Sum Identity for tan ( u + v ) and then derive the tan 2( u ) identity this
way.
If you’re quick with algebra, you may prefer to go in reverse: memorize the Double-Angle Identities, and then guess the Sum Identities.
GROUP 4: POWER-REDUCING IDENTITIES (“PRIs”)
(These are called the “Half-Angle Formulas” in some books.) Memorize: Then,
sin^2 u = 1 ^ cos 2 2 (^ u ) or 12 12 cos 2( u ) tan 2 u = sin^
(^2) u cos 2 u =^
1 cos 2( u )
1 + cos 2( u )
cos 2 u = 1 +^ cos 2 2 (^ u ) or 12 + 12 cos 2( u )
Actually, you just need to memorize one of the sin^2 u or cos 2 u identities and then switch the visible sign to get the other. Think: “sin” is “bad” or “negative”; this is a reminder that the minus sign belongs in the sin^2 u formula.
Obtaining the Power-Reducing Identities from the Double-Angle Identities for cos 2 ( u )
To obtain the identity for sin^2 u , start with Version 2 of the cos 2( u ) identity:
cos 2( u ) = 1 2 sin 2 u
Now, solve for sin^2 u.
2 sin^2 u = 1 cos 2( u )
sin^2 u = 1 ^ cos 2(^ u )
To obtain the identity for cos 2 u , start with Version 3 of the cos 2( u ) identity:
cos 2( u ) = 2 cos 2 u 1
Now, switch sides and solve for cos 2 u.
2 cos 2 u 1 = cos 2( u )
2 cos 2 u = 1 + cos 2( u )
cos 2 u = 1 +^ cos 2(^ u )
GROUP 5: HALF-ANGLE IDENTITIES
Instead of memorizing these outright, it may be easier to derive them from the Power-Reducing Identities (PRIs). We use the substitution = 2 u. (See Obtaining … below.)
The Identities:
sin^ 2
^
= ± 1 ^ cos^ 2
cos^ 2
^
= ± 1 +^ cos^ 2
tan^ 2
^
= ± 1 ^ cos^ 1 + cos
= 1 ^ cos^ sin
= sin^ 1 + cos
For a given , the choices among the ± signs depend on the Quadrant that^ 2
lies in. Here, the ± symbols indicate incomplete knowledge; unlike when we handle the Quadratic Formula, we do not take both signs for any of the above formulas for a given . There are no ± symbols in the last two tan^ 2
^
formulas; there is no problem there of incomplete knowledge regarding signs.
One way to remember the last two tan^ 2
^
formulas: Keep either the numerator or the denominator of the radicand of the first formula, place sin in the other part of the fraction, and remove the radical sign and the ± symbol.
(Chapter 1: Review) 1.46.
Now, 1 cos 0 for all real , and tan^ 2
^
has the same sign as sin (can you see why?), so …
= 1 ^ cos^ sin
To get the third formula, use the numerator’s (instead of the denominator’s) trigonometric conjugate, 1 + cos , when multiplying into the numerator and the denominator of the radicand in the first few steps.
GROUP 6: PRODUCT-TO-SUM IDENTITIES
These can be verified from right-to-left using the Sum and Difference Identities.
The Identities:
sin u sin v = 1 2
cos (^) ( u v ) cos (^) ( u + v )
cos u cos v = 1 2
cos (^) ( u v ) + cos (^) ( u + v )
sin u cos v = 1 2
sin (^) ( u + v ) + sin (^) ( u v )
cos u sin v = 1 2
sin (^) ( u + v ) sin (^) ( u v )
GROUP 7: SUM-TO-PRODUCT IDENTITIES
These can be verified from right-to-left using the Product-To-Sum Identities.
The Identities:
sin x + sin y = 2sin x^ +^ y 2
^
cos x^ ^ y 2
sin x sin y = 2cos x^ +^ y 2
^
sin x^ ^ y 2
cos x + cos y = 2cos x^ +^ y 2
^
cos x^ ^ y 2
cos x cos y = 2sin x^ +^ y 2
^
sin x^ ^ y 2
