Section 7.5 Inverse Trigonometric Functions II Note: A calculator is helpful on some exercises. Bring one to class for this lecture. OBJECTIVE 1: Evaluating composite Functions involving Inverse Trigonometric Funcitons of the Form f f 1 and f 1 f It is imperative that you know and understand the three inverse trigonometric functions introduced in 7.4. A. y = sin 1 x (Say: y is the angle whose sine is x ) 1. Draw the graph of the inverse sine function. 3. The range of the inverse sine function represents an angle whose terminal side lies in B. y = cos 1 x (Say: y is the angle whose cosine is x ) 1. Draw the graph of the inverse cosine function. 3. The range of the inverse cosine function represents an angle whose terminal side lies in C. y = tan 1 x (Say: y is the angle whose tangent is x ) 1. Draw the graph of the inverse tangent function. 3. The range of the inverse tangent function represents an angle whose terminal side lies in
CAUTION: For trigonometric expressions of the form ( f f 1 )(x) or ( f 1 f )(x), the cancellation equations work ONLY if x is in the domain of the inner function. CancellationEquationsforCompositionsofInverseTrigonometricFunctions CancellationEquationsfortheRestrictedSineFunctionanditsInverse forallxintheinterval forall intheinterval. CancellationEquationsfortheRestrictedCosineFunctionanditsInverse forallxintheinterval forall intheinterval. CancellationEquationsfortheRestrictedTangentFunctionanditsInverse forallxintheinterval. forall intheinterval. EXAMPLES: Find the exact value of each expression or state that it does not exist. 7.5.4 7.5.8 7.5.9 7.5.13 7.5.14 7.5.17 7.5.26
OBJECTIVE 2: Evaluating composite Functions involving Inverse Trigonometric Funcitons of the Form f g 1 and f 1 g Method: 1. Evaluate the inner expression and then evaluate the outer expression. 2. It may be necessary to draw a triangle (using x, y, or r) in the appropriate quadrant (depending on if the trig value is positive or negative), determine the value of the missing side and write the trig function requested in the outer expression. 3. If an exact value of the inner expressions cannot be determined, try writing the expressions as an equivalent expression using a cofunction identity. EXAMPLES. : Find the exact value of each expression or state that it does not exist. 7.5.32 7.5.34 7.5.37 7.5.40 7.5.43 7.5.46 OBJECTIVE 3: Functions Understanding the Inverse cosecant, Inverse Secant, and Inverse Cotangent Definition InverseCosecantFunction Theinversecosecantfunction,denotedas,istheinverse of. Thedomainof is andtherangeis.
Definition InverseSecantFunction Theinversesecantfunction,denotedas,istheinverseof. Thedomainof is andtherangeis. Definition InverseCotangentFunction Theinversecotangentfunction,denotedas,istheinverseof. Thedomainof is andtherangeis. EXAMPLES. : Find the exact value of each expression or state that it does not exist. 7.5.48 7.5.49 Most calculators do not have inverse cosecant, inverse secant, or inverse cotangent keys. To use a calculator, it is necessary to rewrite the given inverse cosecant, inverse secant, or inverse cotangent as an expression involving the inverse sine, inverse cosine, or inverse tangent respectively. EXAMPLES. Use a calculator to approximate each value or state that the value does not exist 7.5.51 7.5.52
OBJECTIVE 4: Writing Trigonometric Expressions as Algebraic Expressions Functions In Calculus, it is often necessary to write trigonometric expressions algebraically. In this text u is used as the unknown variable. In calculus x is often (but not always) used. We assume that the variable represents an angle whose terminal side is located in Quadrant I Method: 1. Given the inverse trigonometric expression (inner expression which represents an unknown angle θ ), draw the triangle represented with θ in standard position and the terminal side located in QI 2. Label the given sides of the triangle. Since trigonometric expressions represent ratios of the sides of right triangles, two sides are always given. 3. Determine algebraically the 3 rd side. 4. Write the expression asked for (outside expression). EXAMPLES. Rewrite each trigonometric expression as an algebraic expression involving the variable u. Assume that u > 0 and that the value of the inner trigonometric expression represents an angle θ such that 0 < θ < π 2. 7.5.54 7.5.55 7.5.57