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Ms. June Blackwell's ICM class - 2nd period

6-5 ~ Principal Values of the Inverse Trigonometric Functions

*This web page should teach you how to find principal values of inverse trigonometric functions.

The inverse of a function can be defined by considering only a part of the domain of the function (ie. sine, cos, tan).  The values in the domain of the function are called principal values.  This is how the principal values for sine, cosine, and tangent are defined:

y = Sine x if and only if y = sin x and -90^o < x < 90^o

y = Cos x if and only if y = cos x and 0^o < x < 180^o

y = Tan x if and only if y = tan x and -90^o < x < 90^o

These function definitions allows the range (of the function) to contain all of the possible values of the inverses.  The inverses of the Sine, Cosine, and Tangent functions are called Arcsine, Arccosine, and Arctangent, respectively.  They are defined in the table below.

Okay, now let's try a sample equation!  Find the value.

                    equation:            Arcsin 1/2

        Step 1 (0=theta):            Let 0 = Arcsin 1/2

                         Step 2:            Sin0 = 1/2                                     *(Use Arcsine Function definition

                                                           0 = 30^o                                              from the table above!) :)

                         Step 3:             Therefore, Arcsin 1/2 = 60^o

Now, let's try a word problem a little more advanced.  Use what you know!

You are in ICM doing the "flash problem" with 5 minutes to go!  Lucky for you it is on principal values of the inverse trigonometric functions, your favorite.  It's calculator crunch time!  (Use your calculator if you need to, to solve.)

                                        cos(Sin^-11 - Sin^-1 (¹/₂))

                                        a=Sin^-11  and  b=Sin^-1(¹/₂)

                                        Sin a = 1      Sin b = (¹/₂)  

                                               a = 90^o          b = 30^o

                                        cos(Sin^-11 - Sin^-1(¹/₂)) = cos(a-b)

                                                                                = cos(90^o-30o)

                                                                                = cos(60^o)

                                                                                = ¹/₂

_{Love life, live life, be happy!  Always remember:}

_{"All our dreams can come true, if we have the courage to pursue them."}

_{-Walt Disney}                       

Never forget to live your life the way you want to.  Always have the COURAGE to fight for what you believe in.  Never doubt yourself, and always trust your heart!

*Due credit: Glencoe Advanced Mathematical Concepts and www.cyber-nation.com

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