INVERSE TRIG FUNCTIONS€¦ · INVERSE TRIG FUNCTIONS Section 4.7 Precalculus PreAP/Dual, Revised...
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INVERSE TRIG FUNCTIONS Section 4.7 Precalculus PreAP/Dual, Revised ©2017 [email protected] 8/1/2018 12:52 AM §4.7: Inverse Functions 1
Transcript of INVERSE TRIG FUNCTIONS€¦ · INVERSE TRIG FUNCTIONS Section 4.7 Precalculus PreAP/Dual, Revised...
INVERSE TRIG FUNCTIONS
Section 4.7
Precalculus PreAP/Dual, Revised ©2017
8/1/2018 12:52 AM §4.7: Inverse Functions 1
REVIEW EXAMPLE 1
Determine the inverse of this relation, {(0, –3), (2, 1), and (6, 3)}
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( ) ( ) ( ) 3,0 , 1,2 , 3,6−
REVIEW EXAMPLE 1
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inverse
Mirrored Image
𝒚 = 𝒙
QUESTION
If a function is a relation, is an inverse a function as well?
To find 𝒇−𝟏 (𝒙), we have the HORIZONTAL LINE TEST
inverse
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REVIEW EXAMPLE 2
Determine the inverse of 𝒚 = 𝟑𝒙 − 𝟐
…to Solve the inverse, switch the 𝒙’s and 𝒚’s
y = 3x – 2
= 3 – 2y x
x = 3y – 2( )1 1 2
3 3f x x− = +
x + 2 = 3y
3 2
3 3
y x +=
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INVERSE DEFINITIONS
A. 𝑺𝒊𝒏−𝟏 is read as “the inverse sine”
1. Indicates the inverse of the sine function
2. Cosine’s inverse is 𝑪𝒐𝒔−𝟏, etc…
3. 𝑺𝒊𝒏−𝟏𝒂 = 𝜽, where 𝐒𝐢𝐧 𝜽 = 𝒂
B. There are two ways of writing inverse functions for sine
1. 𝑺𝒊𝒏−𝟏
2. 𝒂𝒓𝒄𝒔𝒊𝒏
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THE INVERSE OF SINE GRAPH
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2
−
2
+
SINE FUNCTION
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THE INVERSE OF SINE
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𝒙 𝒚
𝟎 𝟎
𝝅/𝟐 𝟏
𝝅 𝟎
𝟑𝝅/𝟐 −𝟏
𝟐𝝅 𝟎
𝒙 𝒚
𝟎 𝟎
𝟏 𝝅/𝟐
𝟎 𝝅
−𝟏 𝟑𝝅/𝟐
𝟎 𝟐𝝅
COSINE FUNCTION
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THE INVERSE OF COSINE GRAPH
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THE INVERSE OF COSINE
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𝒙 𝒚
𝟎 𝟏
𝝅/𝟐 𝟎
𝝅 −𝟏
𝟑𝝅/𝟐 𝟎
𝟐𝝅 𝟏
𝒙 𝒚
𝟏 𝟎
𝟎 𝝅/𝟐
−𝟏 𝝅
𝟎 𝟑𝝅/𝟐
𝟏 𝟐𝝅
THE INVERSE OF TANGENT GRAPH
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TANGENT FUNCTION
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Und
Und3
1
3
3
0
3
3−
1−
3−
THE INVERSE OF TANGENT
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𝒙 𝒚
𝟎 𝑼𝒏𝒅
𝝅/𝟐 −𝟏
𝝅 𝟎
𝟑𝝅/𝟐 𝟏
𝟐𝝅 𝑼𝒏𝒅
𝒙 𝒚
𝑼𝒏𝒅 𝟎
−𝟏 𝝅/𝟐
𝟎 𝝅
𝟏 𝟑𝝅/𝟐
𝑼𝒏𝒅 𝟐𝝅
THE INVERSE GRAPH RESTRICTIONS
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A. The restricted domain for sine function is from −𝝅
𝟐,𝝅
𝟐
B. The restricted domain for cosine function is from 𝟎, 𝝅
C. The restricted domain for tangent function is from −𝝅
𝟐,𝝅
𝟐
REVIEW
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Find the exact value of 𝐜𝐨𝐬𝟑𝝅
𝟒
cos =2
2−
3
4
cos =1 Cos− =
EXAMPLE 1
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Find the exact value of 𝐂𝐨𝐬−𝟏 −𝟐
𝟐
2
2
−
3
4
3
4
EXAMPLE 2
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Find the exact value of 𝐚𝐫𝐜𝐬𝐢𝐧𝟏
𝟐
6
YOUR TURN
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Find the exact value of 𝐂𝐨𝐬−𝟏𝟐
𝟐
4
EXAMPLE 3
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Find the exact value of 𝐓𝐚𝐧−𝟏 𝟑
3
EXAMPLE 4
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Find the exact value of 𝐚𝐫𝐜𝐬𝐢𝐧 −𝟑
𝟐
3
−
YOUR TURN
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Find the exact value of 𝐓𝐚𝐧−𝟏 −𝟑
𝟑
6
−
THE INVERSE OF COSECANT GRAPH
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COSECANT FUNCTION
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2 3
3
2
2
und
2−
2−
2 3
3−
1
1
THE INVERSE OF SECANT GRAPH
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SECANT FUNCTION
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2 3
3−
2−
2− und2
2
2 3
3
1− 1
THE INVERSE OF COTANGENT GRAPH
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COTANGENT FUNCTION
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0
0 3
31
3
und
3−
1−
3
3−
und
THE INVERSE RECIPROCAL GRAPHS
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A. The restricted domain for cosecant function is from −𝝅
𝟐,𝝅
𝟐
B. The restricted domain for secant function is from 𝟎, 𝝅
C. The restricted domain for cotangent function is from 𝟎, 𝝅
STEPS
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A. Take the reciprocal of the trig function
B. Take the reciprocal of the answer
C. Switch and solve
cos =1 Sec− =
EXAMPLE 5
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Find the exact value of 𝑺𝒆𝒄−𝟏 − 𝟐
1
2
−
3
4
3
4
EXAMPLE 6
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Find the exact value of 𝐚𝐫𝐜𝐬𝐞𝐜 −𝟐 𝟑
𝟑
5
6
EXAMPLE 7
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Find the exact value of 𝑪𝒐𝒕−𝟏 − 𝟑
5
6
YOUR TURN
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Find the exact value of 𝑪𝒔𝒄−𝟏 − 𝟐
4
−
INVERSES IN THE CALCULATOR
A. Make sure that it is in the appropriate mode
B. It is the 𝑻𝒓𝒊𝒈−𝟏 feature in the graphing calculator
C. For reciprocal functions, type in the 𝑻𝒓𝒊𝒈−𝟏 𝟏
𝒂𝒏𝒔
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EXAMPLE 8
Use the calculator to approximate 𝐚𝐫𝐜𝐭𝐚𝐧(−𝟖. 𝟒𝟓)
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1.4530 −
EXAMPLE 9
Use the calculator to approximate 𝑪𝒐𝒕−𝟏 𝟑
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0.5236
YOUR TURN
Use the calculator to approximate 𝒂𝒓𝒄𝒄𝒔𝒄 −𝟏𝟐
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0.0834 −
ASSIGNMENT
Page 326
5-17 odd (without a calculator), 21-37 EOO (with a calculator), 119-125 all (without), 126-134 all (Round for 4 decimal places)
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