chapter 5 inverse trigonometric functions; trigonometric equations and inequalities
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Chapter 5 Inverse Trigonometric Functions; Trigonometric Equations and Inequalities. 5.1 Inverse sine, cosine, and tangent 5.2 Inverse cotangent, secant, and cosecant 5.3 Trigonometric Equations: An Algebraic Approach 5.4 Trigonometric Equations: A Graphing Calculator Approach. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 5Inverse Trigonometric Functions;
Trigonometric Equations and Inequalities
5.1 Inverse sine, cosine, and tangent 5.2 Inverse cotangent, secant, and cosecant5.3 Trigonometric Equations: An Algebraic Approach5.4 Trigonometric Equations: A Graphing Calculator Approach
5.1 Inverse sine, cosine, and tangent
Inverse sine functionInverse cosine functionInverse tangent function
Inverse Sine Function
Finding the Exact Value of sin-1 x
Example: Find the exact value of sin-1 (√3/2)
Solution:y = sin-1 (√3/2) is equivalent to sin y = √3/2. Find the value of y that lies between –/2 and /2 on the unit circle.
The answer is /3.
Inverse Cosine Function
Finding the Exact Value of cos-1x
Example: Find the exact value of cos-1 ½.
Solution:y = cos-1 ½ is equivalent to
cos y = ½. We find the value of y on the unit circle between 0 and for which this is true.
The answer is /3.
Inverse Tangent Function
Graphs of the tan and tan-1 Functions
Finding the Exact Value of tan-1 x
Example: Find the exact value of tan-1 (-1/√3).
Solution:Y = tan-1 (-1/√3) is
equivalent to tan y = -1/√3. Find the value of y on the unit circle between –/2 and /2 for which this is true.
Answer is –/6.
5.2 Inverse Cotangent, Secant, and Cosecant Functions
Definition of inverse cotangent, secant, and cosecant functions
Calculator evaluation
Domains for Cotangent, Secant and Cosecant
Graphs of Cotangent, Secant, and Cosecant
Finding the Exact Value of arccot (-1)
Example: Find the exact value of arccot (-1)
Solution:y = arccot(-1) is equivalent
to cot y = -1. Find the value of y on the unit circle between 0 and that makes this true.
The answer is 3/4
Identities
5.3 Trigonometric Equations:An Algebraic Approach
IntroductionSolving trigonometric equations using an
algebraic approach
Solving a Simple Sine Equation
Find all solutions in the unit circle to sin x = 1/√2.
Solution:Use the unit circle to
determine that one solution is x = /4.
It can be seen that another point on the circle with the desired height is
x = 3/4.
Suggestions for Solving Trigonometric Equations
Exact Solutions Using Factoring
Example: Find all solutions in [0, 2] to 2 sin2x + sin x = 0
Solution:2 sin2x + sin x = 0sin x(2 sin x + 1) = 0sin x = 0 or sin x = -1/2Find these values on the unit
circle.The solutions are x = 0, ,
7/6, and 11/6.
Exact Solutions Using Identities and FactoringExample: Find all solutions for sin 2x
= sin x, 0 x 2.Solution:sin 2x = sin x2 sin x cos x = sin x2 sin x cos x – sin x = 0sin x (2 cos x – 1) = 0sin x = 0 or cos x = ½From the unit circle we find 4
solutions: x = 0, /3, , and 5/3.
5.4 Trigonometric Equations and Inequalities: A Graphing Calculator ApproachSolving trigonometric equations using a
graphing calculatorSolving trigonometric inequalities using a
graphing calculator
Solutions Using a Graphing Calculator
Example: Graph y1=sin(x/2) and y2= 0.2x – 0.5 over [-4, 4].
Use the INTERSECT command to find that x=5.1609 is the intersection.
Use the ZOOM command to find that there is no intersection in the third quadrant.
Solution Using a Graphing Calculator
Example: Find all real solutions (to four decimal places) to tan(x/2) = 5x – x2 over [0, 3].
Graph y = tan(x/2) and y = 5x – x2 over 0X3 and -10Y10.
Use the INTERSECT command to find three solutions:
x = 0.0000, 2.8292, 5.1272