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