chapter 5.2. 5.2 verifying trigonometric identities in this section you will study techniques for...
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Chapter 5.2
5.2 Verifying Trigonometric IdentitiesIn this section you will study techniques for verifying trigonometric identities.
In 5.3 you will study techniques for solving trigonometric identities
*The key to both is your ability to use the fundamental identities and the rules of algebra to rewrite the trigonometric expressions.
Remember:
•Solving an equation is finding the values that make a statement true.
•For example: is true for
This is an example of a conditional equation.
0sin x nx
5.2 Verifying Trigonometric IdentitiesAn equation that is true for all real values in the domain of the
variable is an identity.
For example: is true for all real numbers x. So, it is an identity.
There is no well-defined set of rules to follow in verifying trigonometric identities, but there are some guidelines.
1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.
2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. (In other words, look for ways to get the expression into something you recognize)
xx 22 cos1sin
5.2 Verifying Trigonometric Identities3. Look for opportunities to use the fundamental identities.
Remember: Sine and cosine pair up well, as do secants and tangents, and cosecants and cotangents.
4. If step 1 through 3 are not helping try converting all terms to sines and cosines.
5. MAKE AN ATTEMPT! Always try something, it might provide insight for you.
Note: Since you are trying to verify one side is the same as the other you can not cross multiply or add the same quantity to both sides. Do not assume both sides are equal, that is what you are trying to verify!
O.K. Let’s get started!
5.2 Verifying Trigonometric Identities
Verify the identity
22
2
sinsec
1sec
2
2
sec
1sec Start with the left side because it is more complicated
2
2
sec
1)1(tan
2
2
sec
tan
)(costan 22
)(coscos
sin 22
2
2sin
Pythagorean identity
Simplify
Reciprocal identity
Quotient identity
Simplify
*Remember*
There can be more than one way to verify an identity
5.2 Verifying Trigonometric IdentitiesTry #5 pg. 3531
222 sin21sincos Verify the identity
5.2 Verifying Trigonometric IdentitiesCombining Fractions Before Using Identities
Verify the identity
2sec2sin1
1
sin1
1
)sin1)(sin1(
sin1sin1
sin1
1
sin1
1
2sin1
2
2cos
2
2sec2
Add fractions
Simplify
Pythagorean identity
Reciprocal identity
5.2 Verifying Trigonometric IdentitiesTry #31 pg. 3542
Verify the identity algebraically0
coscos
sinsin
sinsin
coscos
yx
yx
yx
yx
5.2 Verifying Trigonometric IdentitiesVerify the identity xxx 222 tan)1)(cos1(tan Apply the identities before you multiply
)sin)((sec 22 xx
2
cos
sin
x
x
x
x2
2
cos
sin
x2tan
Reciprocal identity
Rule of exponents
Quotient identity
5.2 Verifying Trigonometric IdentitiesTry #41 pg. 3543
Verify the identity algebraicallyx
x
xxsec
cos
cottan
5.2 Verifying Trigonometric Identities
Verify the identityx
xxx
sin1
costansec
x
x
x
x
x
x
sin1
sin1
sin1
cos
sin1
cos
x
xxx2sin1
sincoscos
x
xxx2cos
sincoscos
x
xx
x
x22 cos
sincos
cos
cos
x
x
x cos
sin
cos
1
xx tansec
Multiply numerator and denominator by (1+sin x)
Multiply
Pythagorean identity
Separate fractions
Simplify
Identities
5.2 Verifying Trigonometric IdentitiesTry #47 pg. 3544
Verify the identity algebraically
sin
cos1
cos1
sin
5.2 Verifying Trigonometric IdentitiesWork with each side separately
Verify the identity
sin
sin1
csc1
cot2
csc1
1csc
csc1
cot 22
csc1
)1)(csc1(csc
1csc
sin
sin
sin
1
sin
sin1
1csc
Left Side Right Side
5.2 Verifying Trigonometric IdentitiesTry #49 pg. 3545
Verify the identity algebraically 1tantan1tan
1tan 23
5.2 Verifying Trigonometric IdentitiesEnriched Pre-Calculus
Verify each identity
xxxxa 2224 tansectantan)
xxxxxb sin)cos(coscossin.) 6443
xxx sincos)cos1( 42
xxx sin)cos(cos 64
)1(sectan 22 xx
xxx 222 tansectan
))(tan(tantan 224 xxx
xxxxx sincossincossin 4243
Rewrite as separate factors
Rewrite as separate factors
Pythagorean identity
Pythagorean identity
Multiply
Multiply
5.2 Verifying Trigonometric Identities
Verify the identity
Try #63 pg. 3556
xxxx 3235 tansectantan
THE END