Trig Review

1
Trig Review

• AP Calculus
• 2008/2009 School Year

2
Two Historical Perspectives

• The Unit Circle
• Right Triangle Trigonometry

3
The Unit Circle

• x2y21
• Wrap the real number line around the circle
  counterclockwise so that zero is matched with the
  point (0,1).
• Circumference of a circle is 2p, so the point
  (0,1) is also matched with 2p, 4p, etc.
• The point (0,1) is matched with p/2 (-1,0) is
  matched with p (0,-1) is matched with (3p)/2
  etc.
• Unit of measurement called radians.
• Each radian measure corresponds to the arc length
  from zero to that point on the unit circle.

4
Six Trig Functions

• The coordinates (x,y) of a point of the unit
  circle are functions of ?.
• Use the coordinates (x,y) to define six trig
  functions

sin ? y cos ? x tan ? y/x
csc ? 1/y sec ? 1/x cot ? x/y

• Functions on the right are reciprocals of
  functions on the left.
• tan ? sin ? / cos ?

• cot ? cos ? / sin ?

5
Right Triangle Trig

• Six trig functions of the angle ? can also be
  defined as ratios of the sides of right triangle,
  where ? is an acute angle of the triangle.

sin ? opp/hyp cos ? adj/hyp tan ?
opp/adj csc ? hyp/opp sec ? hyp/adj cot ?
adj/opp
hypotenuse
opposite
?
adjacent
6
Sines, Cosines, and Tangents of Special Angles
7
Reference Angles

• Angles on previous page called reference angles
• Memorize sine and cosine of reference angles.
• Use tan ? sin ? / cos ? and reciprocals to find
  other trig functions of the reference angles.
• If ? is in the 2nd, 3rd, or 4th quadrant, pick an
  appropriate reference angle to find value of
  function and then add the correct sign.

8
Domain and Range
9
Graphs of Trig Functions

See graphing calculator
10
Transforming the Graphs

• Amplitude a
• Period (2p/b)
• Horizontal translation c
• Vertical translation d

11
Trigonometric Identities

• Pythagorean Identities
• sin2 x cos2 x 1
• Divide both sides by cos2 x 1 tan2 x sec2 x
• Divide both sides by sin2 x 1 cot2 x csc2 x
• Other identities arent that important
• You can verify trig identities, but people in the
  real world dont do that.

12
Solving Trig Equations

• Basic trig equation (collecting like terms,
  extracting square roots, factoring)
• Equations of a quadratic type
• Rewriting with a single trig function
• Functions of multiple angles
• See practice problems handout for practice. These
  types of questions are important for Calculus!

13
Other Topics

• Sum and difference formulas
• Multiple-angle and product to sum formulas
• Laws of Sines and Cosines
• Inverse trig functions
• We will review inverse trig functions in class.
  The other topics above are not essential for
  Calculus.