Precalc 6'1

1
Precalc 6.1

• Right Triangle Trigonometry. And the left angles
  who love them.

2
INTRO

• Remember geometry? No. Well, its back. With a
  vengeance.
• Angle Set of points defined by two rays sharing
  an endpoint?
• In trig, we often interpret angles as rotations
  of rays. For example, we can start with ray l1
  having endpoint O and rotate it to create ray
  l2.?
• Names?l1 initial side
• O vertex?l2 terminal side

3
Angular

• We can measure angles in degrees.?
• One full revolution 360.
• Note we often use lowercase Greek letters to
  represent angles Commonly used a, ?, ?, ?, ?
• Counterclockwise rotation positive angle (i.e.
  90)?
• Clockwise rotation negative angle (I.e. -270)

4
DMS

• Sometimes we need measurements smaller than a
  degree. ?
• Minute (1/60) Second (1/3600)
• Notation ? 732741
• Theta is 73 degrees, 27 minutes and 41 seconds.
• Helpful conversion 90 895960

5
Come and be converted.

• Write 351527 in decimal form.
• Solution 35 15(1/60) 27(1/3600)
  35.2575
• Write 48.3625 in DMS form.
• Solution The 48 is already OK.
• For minutes
• For seconds
• Answer 482145

6
You do it.

• Convert 91.8725 into DMS
• Solution 915221
• The ratios of the sides of a right triangle are
  special indeed. They form several key buttons on
  your calculator with exotic names such as sin and
  tan - both things you like to do over the summer.
  

7
Rt ?
Heres a spiffy right triangle with legs a b
with a hypotenuse c. Ive also named the angles
a and ?.

• Q What are the six possible ratios I could
  create with the sides??
• A a/c b/c a/b b/a c/b c/a
• These ratios have special names that are related
  to the interior angles a and ?.??
• sine a a/c cosine a b/c tan a
  a/b
• cosecant a c/a secant a c/b ???cotan a b/a

8
Wheres my keys?

• Q So how can we remember all of this??
• A Lets revisit our right triangle and name it.
• Think SOHCAHTOA! lt---Our trigonometric princess.?
• Sine Opposite Hypotenuse??
• Cosine Adjacent Hypotenuse??
• Tangent Opposite Adjacent

OPP Opposite Side (to angle) ADJ Adjacent
Side (to angle) HYP Hypotenuse
9
OK, but what about secant, cosecant, and
cotangent?

• Check it out
• cosecant a 1/sin a csc a
• secant a 1/cos a sec a
• cotangent a 1/tan a cot a
• I know it SEEMS like COsecant should go with
  COsine. But remember it this way - opposites
  attract!?

10
Show me da ?

• Given the ?shown, find sin, cos, tan, csc, sec,
  cot of ?.

sin ? 5/13 csc ? 13/5 cos ? 12/13 sec ?
13/12 tan ? 5/12 cot ? 12/5
You might wonder, OK, I know the trig ratios,
but whats ?? This is an excellent question that
will be answered soon
11
Greatest. Trick. Ever.

• Suppose I want to know what the sin of 30 is.
  You could type it on calc (assuming your calc is
  in degree mode) but the answer will be a decimal
  approximation. Is there an exact answer? You
  betcha! What Im about to show you is the
  greatest trig reference tool of all time.

12
Exact Trig Values of Common Angles.
sin
cos

• STEPS
• Create the fractions over 2.
• Number from 0 to 4 and then 4 to 0.
• Put square roots on all of the numerators

v
v
0
/2
/2
4
0
v
v
30
/2
/2
1
3
v
v
45
/2
/2
2
2
v
v
60
/2
/2
3
1
v
v
/2
/2
4
90
0
Is there a trick for tangent? You bet! tangent
sin/cos! Booyahbase!