Solving Trigonometric Functions

Importance

• Key concept in IV math exams.
• Familiarity with solving trig functions is essential.

Example Problem

• Solve ( \sin(X) = \frac{1}{2} )
• Recognize ( \sin ), ( \cos ), and ( \tan ) ratios for specific angles.

Magic Triangle

• Useful for solving without a calculator.
• Triangle with sides 1, 2, ( \sqrt{3} ):
  • 30 degrees (( \pi/6 ))
  • 60 degrees (( \pi/3 ))
• ( \sin ) is opposite over hypotenuse.

Determining the Angle

• Identify the angle using known ratios.
• ( \sin(\pi/6) = \frac{1}{2} )
• The angle for ( \sin(X) = \frac{1}{2} ) is ( X = \pi/6 ).

Tool Angle

• ( X = \pi/6 ) is the tool angle.
• Use the tool angle to find all solutions within a given domain, e.g., ( 0 \leq X \leq 2\pi ).

Unit Circle

• Use to determine which quadrants have positive ( \sin ) values.
• Positive ( \sin ) quadrants: S (2nd) and A (1st).

Solutions in Given Domain

• Within ( 0 ) to ( 2\pi ):
  • ( X = \pi/6 )
  • ( X = 5\pi/6 )
• No solutions in 3rd and 4th quadrants (negative ( \sin ) values).

Graphical Representation

• Graph of ( \sin(X) ) function from ( 0 ) to ( 2\pi ).
• ( \sin(X) = \frac{1}{2} ) at two points in this range.

Solving a More Complex Example

• Example: ( 2\cos(X) + 3 = 2 )
  • Rearrange to ( \cos(X) = -\frac{1}{2} )
  • Find angle where ( \cos(\text{angle}) = \frac{1}{2} ), which is 60 degrees (( \pi/3 )).
  • Adjust for negative value: focus on negative ( \cos ) quadrants.

Practice

• Practice rearranging equations to isolate trig functions.
• Solve for multiple angles within specified domains.

Good luck with practicing solving trigonometric functions!