What is a FUNCTION? an input/output machine where…

• …each input has EXACTLY one output.

• …outputs are 100% predictable.

Example:

So what does an INVERSE function do?

• It reverses the process of a function.

• Consequently, it changes the outputs BACK to inputs.

INVERSE example:

INVERSE FunctionMachine

What about trigonometric functions?

• Sine, cosine, and tangent have ANGLES as inputs.

• Their outputs are a ratio of sides (Remember Soh-Cah-Toa)

𝜃𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒sin( )𝜃

How do INVERSE trigonometric functions work?

• The input of an INVERSE trig function is a ratio of sides.

• The output of an INVERSE trig function is an ANGLE.

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒sin-1( ) 𝜃

𝑜𝑝𝑝

ℎ𝑦𝑝

tan−1𝑜𝑝𝑝

𝑎𝑑𝑗= 𝜃

cos−1𝑎𝑑𝑗

ℎ𝑦𝑝= 𝜃

sin−1𝑜𝑝𝑝

ℎ𝑦𝑝= 𝜃𝑠𝑖𝑛 𝜃 =

𝑜𝑝𝑝

ℎ𝑦𝑝

𝑐𝑜𝑠 𝜃 =𝑎𝑑𝑗

ℎ𝑦𝑝

𝑡𝑎𝑛 𝜃 =𝑜𝑝𝑝

𝑎𝑑𝑗

FUNCTION INVERSE FUNCTION

Choose the OUTPUT…

𝜶

𝜷

B

AC

sin 𝛼 =

BC AB

Choose the OUTPUT…

𝜶

𝜷

B

AC

sin 𝛽 =

CA

AB

Choose the OUTPUT…

𝜶

𝜷

B

AC

sin−1𝐴𝐶

𝐴𝐵=

𝜷

Choose the OUTPUT…

𝜶

𝜷

B

AC

tan−1𝐴𝐶

𝐶𝐵=

𝜷