Analytic Trigonometry. Chapter 6. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A. The Inverse Sine, Cosine, and Tangent Functions. Section 6.1. One-to-One Functions.
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Analytic Trigonometry Chapter 6 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAA
One-to-One Functions • A one-to-one function is a function f such that any two different inputs give two different outputs • Satisfies the horizontal line test • Functions may be made one-to-one by restricting the domain
Inverse Functions • Inverse Function: Function f {1 which undoes the operation of a one-to-one function f.
Inverse Functions • For every x in the domain of f, f {1(f(x)) = x and for every x in the domain of f {1, f(f {1(x)) = x • Domain of f = range of f {1, and range of f = domain of f {1 • Graphs of f and f {1, are symmetric with respect to the line y = x • If y = f(x) has an inverse, it can be found by solving x = f(y) for y. Solution is y = f {1(x) • More information in Section 4.2
Inverse Sine Function • Inverse sine function: Inverse of the domain-restricted sine function
Inverse Sine Function y = sin{1x means x = sin y • Must have {1 ·x· 1 and • Many books write y = arcsin x WARNING! • The {1 is not an exponent, but an indication of an inverse function • Domain is {1 ·x· 1 • Range is
Approximate Values of the Inverse Sine Function • Example. Find approximate values of the following. Express the answer in radians rounded to two decimal places. (a) Problem: Answer: (b) Problem: Answer:
Inverse Cosine Function • Inverse cosine function: Inverse of the domain-restricted cosine function
Inverse Cosine Function y = cos{1x means x = cos y • Must have {1 ·x· 1 and 0 ·y·¼ • Can also write y = arccos x • Domain is {1 ·x· 1 • Range is 0 ·y·¼
Approximate Values of the Inverse Cosine Function • Example. Find approximate values of the following. Express the answer in radians rounded to two decimal places. (a) Problem: Answer: (b) Problem: Answer:
Identities • Two functions f and g are identically equal provided f(x) = g(x) for all x for which both functions are defined • The equation above f(x) = g(x) is called an identity • Conditional equation: An equation which is not an identity
Simplifying Using Identities • Example. Simplify the following expressions. (a) Problem: cot µ¢ tan µ Answer: (b) Problem: Answer:
Establishing Identities • Example. Establish the following identities (a) Problem: (b) Problem:
Guidelines for Establishing Identities • Usually start with side containing more complicated expression • Rewrite sum or difference of quotients in terms of a single quotient (common denominator) • Think about rewriting one side in terms of sines and cosines • Keep your goal in mind – manipulate one side to look like the other
Key Points • Identities • Fundamental Trigonometric Identities • Simplifying Using Identities • Establishing Identities • Guidelines for Establishing Identities
Sum and Difference Formulas for Cosines • Theorem. [Sum and Difference Formulas for Cosines] cos(® + ¯) = cos ® cos ¯ { sin ® sin ¯ cos(® { ¯) = cos ® cos ¯ + sin ® sin ¯
Sum and Difference Formulas for Sines • Theorem. [Sum and Difference Formulas for Sines] sin(® + ¯) = sin ® cos ¯ + cos ® sin ¯ sin(® { ¯) = sin ® cos ¯ { cos ® sin ¯
Sum and Difference Formulas for Sines • Example. If it is known that and that find the exact values of: (a) Problem: cos(µ + Á) Answer: (b) Problem: sin(µ { Á) Answer:
Key Points • Sum and Difference Formulas for Cosines • Identities Using Sum and Difference Formulas • Sum and Difference Formulas for Sines • Sum and Difference Formulas for Tangents • Sum and Difference Formulas With Inverse Functions
