1) A function is an input/output machine where each input has exactly one output and the outputs are 100% predictable. 2) An inverse function reverses the process of a function by changing the outputs back to inputs. 3) Trigonometric functions like sine, cosine, and tangent take angles as inputs and output ratios of sides, while inverse trigonometric functions take ratios of sides as inputs and output angles.

1. What is a FUNCTION?
 an input/output machine where…
• …each input has
EXACTLY one output.
• …outputs are 100%
predictable.

2. Example:

3. So what does an
INVERSE function do?
• It reverses the process of
a function.
• Consequently, it changes
the outputs BACK to
inputs.

4. INVERSE example:
INVERSE
Function
Machine

5. What about trigonometric
functions?
• Sine, cosine, and tangent have
ANGLES as inputs.
• Their outputs are a ratio of sides
(Remember Soh-Cah-Toa)

6. 𝜃
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sin( )𝜃

7. 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.

8. 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sin-1( ) 𝜃
𝑜𝑝𝑝
ℎ𝑦𝑝

9. tan−1
𝑜𝑝𝑝
𝑎𝑑𝑗
= 𝜃
cos−1
𝑎𝑑𝑗
ℎ𝑦𝑝
= 𝜃
sin−1
𝑜𝑝𝑝
ℎ𝑦𝑝
= 𝜃𝑠𝑖𝑛 𝜃 =
𝑜𝑝𝑝
ℎ𝑦𝑝
𝑐𝑜𝑠 𝜃 =
𝑎𝑑𝑗
ℎ𝑦𝑝
𝑡𝑎𝑛 𝜃 =
𝑜𝑝𝑝
𝑎𝑑𝑗
FUNCTION INVERSE FUNCTION

10. Choose the OUTPUT…
𝜶
𝜷
B
AC
sin 𝛼 =
BC AB

11. Choose the OUTPUT…
𝜶
𝜷
B
AC
sin 𝛽 =
CA
AB

12. Choose the OUTPUT…
𝜶
𝜷
B
AC
sin−1
𝐴𝐶
𝐴𝐵
=
𝜷

13. Choose the OUTPUT…
𝜶
𝜷
B
AC
tan−1
𝐴𝐶
𝐶𝐵
=
𝜷