Trigonometry Chapter 5 Lecture Notes Section 5.1 ... Resources files/LHS Trig 8th... ·...
Transcript of Trigonometry Chapter 5 Lecture Notes Section 5.1 ... Resources files/LHS Trig 8th... ·...
LHS Trig 8th ed Ch 5 Notes F07 O’Brien
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Trigonometry Chapter 5 Lecture Notes
Section 5.1 Fundamental Identities
I. Negative-Angle Identities
sin (– θ) = – sin θ csc (– θ) = – csc θ tan (– θ) = – tan θ
cot (– θ) = – cot θ cos (– θ) = cos θ sec (– θ) = sec θ
One of the easiest ways to remember the negative-angle identities is to remember that only cosine and its reciprocal, secant are even functions. For even functions, f(– x) = f(x) which means these functions have y-axis symmetry. The other four trig functions (sine, cosecant, tangent, and cotangent) are odd functions. For odd functions, f(– x) = – f(x) which means these functions have origin symmetry. Example 1 Since tangent is an odd function, if tan x = 2.6, then tan (–x) = –2.6. (#1)
II. Reciprocal Identities
θsin
1θ csc = θcos
1θ sec = θtan
1θcot =
III. Quotient Identities
θ cosθsin θtan =
θsin θ cosθcot =
IV. Pythagorean Identities
1θcosθsin 22 =+ θsec1θtan 22 =+ θcscθcot1 22 =+
V. Using the Fundamental Identities
A. Finding Trigonometric Function Values Given One Value and the Quadrant
Example 2 Given 31 cot x −= and x is in quadrant IV, find sin x. (modified #6)
xcot1 x csc x cot1 xcsc 222 +±=→+= ; cosecant and sine are negative in IV;
310
910
311 x csc
2−=−=⎟
⎠⎞
⎜⎝⎛−+−= →
10103
103
xcsc1sin x −=−==
Now find the three remaining trigonometric functions of x.
3cot x
1tan x −==
sin xcot x xcos ⋅= → 1010
10103
31 xcos =−⋅−=
10 xcos
1 xsec ==
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B. Using Identities to Rewrite Functions and Expressions
Example 3 Use identities to rewrite cot x in terms of sin x. (#44)
sin x
xcoscot x = and from 1xcosxsin 22 =+ we know xsin1 xcos 2−±=
therefore, sin x
xsin1cot x2−±
= .
Example 4 Rewrite the expression θsin θcot θ sec ⋅⋅ in terms of sine and cosine and simplify. (#50)
1θsin θcosθsin θ cosθsin
θsin θ cos
θcos1θsin θcot θ sec =
⋅⋅
=⋅⋅=⋅⋅
Example 5 Use identities to rewrite the expression sin2x + tan2x + cos2x in terms of sec x. (#63)
sin2x + cos2x = 1, so sin2x + tan2x + cos2x = 1 + tan2x which equals sec2x
************************************************************************************ Section 5.2 Verifying Trigonometric Identities
I. Verifying Trigonometric Identities
A. An identity is an equation that is true for all of its domain values.
B. To verify an identity, we show that one side of the identity can be rewritten to look exactly like the other side.
C. Verifying identities is not the same as solving equations. Techniques used in solving equations, such as adding the same term to both sides or multiplying both sides by the same factor, are not valid when verifying identities. II. Hints for Verifying Trigonometric Identities
A. Know the fundamental identities and their equivalent forms inside out and upside down.
Example 1 sin2x + cos2x = 1 is equivalent to cos2x = 1 – sin2x
Example 2 xcos
sin xtan x = is equivalent to tan x xcossin x ⋅=
B. Start working with the more complicated side of the identity and try to turn it into the simpler side. Do not work on both sides of the identity simultaneously.
C. Perform any indicated operations such as factoring, squaring binomials, distributing, or adding fractions.
Example 3 1sin3sin2 2 ++ x x can be factored to ( )( )1sin1sin2 ++ xx (#17)
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Example 4 xx cos
1sin
1+ can be added by getting a common denominator
xxxx
xx
xxx
xxx cossinsincos
sinsin
cos1
coscos
sin1
cos1
sin1
⋅+
=⋅+⋅=+
D. Sometimes it is helpful to express all trigonometric functions on one side of an identity in terms of sine and cosine.
Example 5 Verify xxx sin
sectan
= (#34)
xxxx
x
xx
xx sin
1cos
cossin
cos1
cossin
sectan
=⋅==
E. Fractions with a sum in the numerator and a single term in the denominator can be rewritten as the sum of two fractions.
Example 6 Verify xxx
x cotcscsin
cos1+=
+
xxxx
xxx cotcsc
sincos
sin1
sincos1
+=+=+
Fractions with a difference in the numerator and a single term in the denominator can be rewritten as the difference of two fractions.
F. Sometimes it is helpful to rewrite one side of the identity in terms of a single trigonometric function.
Example 7 Verify xxxx cossec
cossin 2
−= (#42)
xxxx
xxx
xx cossec
coscos
cos1
coscos1
cossin 222
−=−=−
=
G. Multiplying both the numerator and denominator of a fraction by the same factor (usually the conjugate of the numerator or denominator) may yield a Pythagorean identity and bring you closer to your goal.
Example 8 Verify xx
x 2cossin1
sin11 −
=+
.
xx
xx
xx
xx 22 cossin1
sin1sin1
sin1sin1
sin11
sin11 −
=−
−=
−−
⋅+
=+
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H. As you selection substitutions, keep in mind the side you are not changing. It represents your goal. Look for the identity or function which best links the two sides.
Example 9 Verify ( )x
xx 222
sin11cot1tan
−=+ .
( )xx
xxx
xxx 2222
2222
sin11
cos1sec1tan
tan11tancot1tan
−===+=⎟
⎠
⎞⎜⎝
⎛ +=+
I. If you get really stuck, abandon the side you’re working on and start working on the other side. Try to make the two sides “meet in the middle.”
Example 10 ( )xxxx
sin1sin1tansec 2
+−
=−
working on left side: working on right side:________
( ) =− 2tansec xx =+−
xx
sin1sin1
=+− xxxx 22 tantansec2sec =−−
⋅+−
xx
xx
sin1sin1
sin1sin1
=+⋅−xx
xx
xx 2
2
2 cossin
cossin
cos12
cos1 =
−+−
xxx
2
2
sin1sinsin21
xx
xx
x 2
2
22 cossin
cossin2
cos1
+− =+−
xxx
2
2
cossinsin21
xx
xx
x 2
2
22 cossin
cossin2
cos1
+−
************************************************************************************ Section 5.3 Sum and Difference Identities for Cosine
I. Cofunction Identities
cos (90° – θ) = sin θ sin (90° – θ) = cos θ
cot (90° – θ) = tan θ tan (90° – θ) = cot θ
csc (90° – θ) = sec θ sec (90° – θ) = csc θ
Note: The angles θ and 90° – θ can be negative and / or obtuse.
Example 1
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Example 2
Example 3
II. Sum and Difference Identities for Cosine
cos (A + B) = cos A cos B – sin A sin B [Functions stay together, operator changes.]
cos (A – B) = cos A cos B + sin A sin B [Functions stay together, operator changes.]
Example 4
III. Applying the Sum and Difference Identities
A. Reducing cos (A – B) to a Function of a Single Variable
Example 5
B. Finding cos (s + t) Given Information about s and t
Example 6
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C. Verification of an Identity
Example 7
************************************************************************************ Section 5.4 Sum and Difference Identities for Sine and Tangent
I. Sum and Difference Identities for Sine
sin (A + B) = sin A cos B + cos A sin B [Functions mix; sign stays.]
sin (A – B) = sin A cos B – cos A sin B [Functions mix; sign stays.]
II. Sum and Difference Identities for Tangent
( )tanAtanB1
tanBtanABAtan−
+=+ ( )
tanAtanB1tanBtanABAtan
+−
=−
III. Applying the Sum and Difference Identities
A. Finding Exact Sine and Tangent Function Values
Example 1
Example 2
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Example 3
B. Writing Functions as Expressions Involving Functions of θ
Example 4
Example 5
C. Finding Function Values and the Quadrant of A + B
Example 6
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D. Verifying an Identity Using Sum and Difference Identities
Example 7
************************************************************************************ Section 5.5 Double-Angle Identities
I. Double-Angle Identities
cos(A)(A)sin 2(2A)sin ⋅= (A)tan1
(A) tan 2(2A)tan 2−=
(A)sin(A)cos(2A) cos 22 −= 1(A)cos 2(2A) cos 2 −= (A)sin 21(2A) cos 2−=
A. Finding Function Values of θ Given Information about 2θ
Example 1
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B. Finding Function Values of 2θ Given Information about θ
Example 2
C. Using an Identity to Write an Expression as a Single Function Value or Number
Example 3
Example 4
D. Verifying a Double-Angle Identity
Example 5
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E. Deriving a Multiple-Angle Identity
Example 5
II. Product-to-Sum Identities
( ) ( )[ ]BA cosBA cos21B cosA cos −++=⋅ ( ) ( )[ ]BA cosBA cos
21Bsin Asin +−−=⋅
( ) ( )[ ]BAsin BAsin 21B cosAsin −++=⋅ ( ) ( )[ ]BAsin BAsin
21Bsin A cos −−+=⋅
Using a Product-to-Sum Identity
Example 6
III. Sum-to-Product Identities
⎟⎠⎞
⎜⎝⎛ −
⋅⎟⎠⎞
⎜⎝⎛ +
=+2
BAcos2
BAsin 2Bsin Asin ⎟⎠⎞
⎜⎝⎛ −
⋅⎟⎠⎞
⎜⎝⎛ +
=−2
BAsin2
BAcos 2Bsin Asin
⎟⎠⎞
⎜⎝⎛ −
⋅⎟⎠⎞
⎜⎝⎛ +
=+2
BAcos2
BAcos 2B cosA cos ⎟⎠⎞
⎜⎝⎛ −
⋅⎟⎠⎞
⎜⎝⎛ +
−=−2
BAsin2
BAsin 2B cosA cos
Using a Sum-to-Product Identity
Example 7
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************************************************************************************ Section 5.6 Half-Angle Identities
I. Half-Angle Identities
2
A cos12Acos +
±= 2
A cos12Asin −
±= A cos1A cos1
2Atan
+−
±=
A cos1
Asin 2Atan
+=
Asin A cos1
2Atan −=
In the first three half-angle identities, the sign is chosen based on the quadrant of 2A .
II. Applying the Half-Angle Identities
A. Using a Half-Angle Identity to Find an Exact Value
Example 1
B. Finding Function Values of 2θ Given Information about θ
Example 2
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C. Finding Function Values of θ Given Information about 2θ
Example 3
D. Using an Identity to Write an Expression as a Single Trigonometric Function
Example 4
Example 5
E. Verifying an Identity
Example 6