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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
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.
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.
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
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
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.
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 Factoring
Example: 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 Approach
Solving trigonometric equations using a graphing calculator
Solving 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.