using fundamental identities objectives: 1.recognize and write the fundamental trigonometric...
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Using Fundamental Identities
Objectives:
1.Recognize and write the fundamental trigonometric identities2.Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions
WHY???Fundamental trigonometric identities can be used to simplify trigonometric expressions, such as for the coefficient of friction.
Fundamental Trigonometric Identities
Reciprocal Identities
uu
uu
uu
uu
uu
u
tan
1cot
cos
1sec
sin
1csc
cot
1tan
sec
1cos
csc
1sin
Quotient Identities
tanusinu
cosu cot u
cosu
sinu
Fundamental Trigonometric Identities
Pythagorean Identities
sin2 u cos2 u1 1 tan2 usec2 u
1 cot2 ucsc2 u
Even/Odd Identities
sin( u) sinu cos( u) cosu tan( u) tanu
csc( u) cscu sec( u) secu cot( u) cot(u)
Fundamental Trigonometric Identities
Cofunction Identities
sin2
u
= cosu cos
2
u
sinu
tan2
u
cot u cot
2
u
tanu
sec2
u
cscu csc
2
u
secu
Example: If and Ө is in quadrant II, find each function value.
a) sec Ө
To find the value of this function, look for an identity that relates tangent and secant.
Tip: Use Pythagorean Identities.
b) sin Ө
7
c) cot ( Ө )
Example: If and Ө is in quadrant II, find each function value.
(Cont.)Tip: Use Quotient Identities. Tip: Use Reciprocal and
Negative-Angle
Identities.
2. Use the values
cos x > 0 and identities to find the values of all six trigonometric functions.
1sinx and
2
What quadrant will you use? 1st quadrant
1cscx
sinx 1
1/ 2 2
2 2sin x cos x 1 2
2 1cos x 1
2
11
4 3
4
3cosx
2
1secx
cosx 2
3 2 3
3
sinxtanx
cosx
123
2
1
3
33
1cot x
tanx 3
1 3
Using Identities to Evaluate a Function
Use the given values to evaluate the remaining trigonometric functions
(You can also draw a right triangle)
secu3
2,tanu 0
csc 5,cos 0
tan x 33
,cos x 32
Simplify an ExpressionSimplify an Expression Simplify cot x cos x + sin x. Click for answer.
x
xx
sin
coscot
xx
xxx
x
xsin
sin
cossincos
sin
cos 2
xxx
xxcsc
sin
1
sin
sincos 22
2Simplif y cos xcscx cscx
Example: Simplify
1. Factor csc x out of the expression.
2cscx cos x 1
2. Use Pythagorean identities to simplify the expression in the parentheses.
2cscx cos x 1
2 2sin x cos x 1 2 2sin x cos x 1
2cscx sin x
3. Use Reciprocal identities to simplify the expression.
2cscx sin x
21sin x
sinx
2sin xsinx
sinx
Simplifying a Trigonometric Expression
sin x cos2 x sin x
sec2 x(1 sin2 x)
tan2 x
sec2 x
Factoring Trigonometric Expressions
sec2 1-Factor the same way you would factor any quadratic.- If it helps replace the “trig” word with x
-Factor the same way you would factor
sec2 1
x 2 1
x 2 1(x 1)(x 1) so sec2 (sec 1)(sec 1)
2b. 2csc x 7cscx 6
Make it an easier problem.Let a = csc x
2a2 – 7a + 6(2a – 3)(a – 2)
Now substitute csc x for a. 2cscx 3 cscx 2
2Factor sec x 3tanx 1.
1. Use Pythagorean identities to get one trigonometric function in the expression.2 2sec x tan x 1.
2tan x 1 3tanx 1
2tan x 3tanx 2
2. Now factor.
tanx 2 tanx 1
Factoring Trigonometric Expressions
4 tan2 tan 3
csc2 x cot x 3
More Factoring
sin2 x csc2 x sin2 x
1 2cos2 x cos4 x
sec3 x sec2 x sec x 1
Adding Trigonometric Expressions (Common Denominator)
sin1 cos
cossin
sinsin
sin
1 cos
cossin
(1 cos)
(1 cos)
(sin)(sin) (cos)(1 cos)
(1 cos)(sin)
sin2 cos cos2(1 cos)(sin)
1 cos
(1 cos)(sin)
1
sin
csc
sin2 cos2 1
Adding Trigonometric Expressions
1
sec x 1 1
sec x 1
tan x sec2 x
tan x
Rewriting a Trigonometric Expression so it is not in Fractional Form
1
1 sin x
5
tan x sec x
tan2 x
csc x 1
Trigonometric Substitution
4 x 2
x 2tan
64 16x 2
x 2cos
x 2 4
x 2sec
x 2 100
x 10tan