Ms. Kresovic’s Adv Pre Calc

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Review:

  1. The 30-60-90 & 45-45-90 triangles where the hypotenuse = 1 unit.


  1. The Unit Circle (the 1st Quad.)

Ms. Kresovic’s Adv Pre Calc

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4.3: Right Triangle Trigonometry

3. Definitions of Trigonometric Functions

wrt triangle sides / wrt any circle / wrt Unit Circle, r = 1
sin = / /
cos = / /
tan = / /
wrt triangle sides / wrt any circle / wrt Unit Circle, r = 1
csc =
sec  =
cot =

*WRT: with respect to

Ms. Kresovic’s Adv Pre Calc

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  1. What are the exact 1st Quad values? Think about the unit circle.

Angle / Degrees / 0 / 30 / 45 / 60 / 90
Radians () / 0 / /6 / /4 / /3 / /2
Rad. (deci.) / 0.0000 / 0.5236 / 0.7854 / 1.0472 / 1.5708
sin = / 0 / ½ / / / 1
cos = / 1 / / / ½ / 0
tan = / 0 / / 1 / / Und
csc = / Und / 2 / / / 1
sec  = / 1 / / / 2 / Und
cot = / Und / / 1 / / 0

How would these values transfer to the Cartesian plane?

Graph of sin(x) and csc(x) , D: [–2pi, 2pi], R: [-2, 2]

Graph of cos(x) and sec(x)

Graph of tan(x) and cot(x)

5. Identities (some of them)

Reciprocal (a = 1/a)

Quotient (written in cosine and sine)

Pythagorean Identities (In any right triangle, leg2 + leg2 = hypotenuse2. Now put the right triangle in the unit circle, such that the hypotenuse = radius = 1.)

Ms. Kresovic’s Adv Pre Calc

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Ms. Kresovic’s Adv Pre Calc

Page 1 of 6

Now what does the equation look like if you divide both sides by sine2?

What does the original equation look like if you divide both sides by cosine2?

Ms. Kresovic’s Adv Pre Calc

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Example of Pyth Id for any angle:

My Birthday is 7.02. This value is bigger than 6.28 or 2, so this angle goes around the circle and stops at 7.02 – 6.28 or 0.74 radians. If I divide this by pi I can rewrite the angle as 0.2355 (so that it will be easier to graph on the SketchPad graph shown below.

Cos2(0.2355) + Sin2(0.2355) = 1

(0.7071)2 + (0.6594) 2 =1.0000

Now use your birthday. If you would rather work in degrees, then take your birthday and multiply by 180 / pi. The circle below is marked for all multiples of 30 and 45.

Objective: Students will empirically observe that the Pythagorean Identity holds true for any angle.

Use your birthday for the angle :______

Find and label this point on the circle. Draw a perpendicular to the horizontal axis and then your triangle. Use the x and y values, cosine and sine respectively, to show

Cos2() + Sin2() = 1

Ms. Kresovic’s Adv Pre Calc

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Verifying trig statements using identities (good work-out exam exercises).

34. csc tan = sec 35. tan cos = sin

37. (1 + cos)(1  cos) = sin2

38. (csc + cot)(csc cot) = 1

40. (tan + cot)/tan = csc2

Transfer the concepts from 4.3 to any quadrant.

4.4: Trig functions of any angle

Definition of Reference Angle: If the angle is in standard position (the initial side is the positive x-axis and the vertex is at the origin), the reference angle is the acute angle formed with the terminal side and the horizontal axis.

Text exercises (p284)

Directions: State the Quadrant in which the angle lies

13. sin  < 0 and cos  < 0 ______

14. sec  > 0 and cot  < 0 ______

15. cot  > 0 and cos  > 0 ______

16. tan  > 0 and csc  < 0 ______

HOMEWORK (all due next class)

4.3: 274/ 6 – 54 multiples of 6 (9 exercises) and

4.4: 284/ 6 – 96 multiples of 6 (16 exercises).

Total of 25 exercises.

Read 4.5 & 4.6 (graphing trig functions).

We will complete these 2 sections next class.

Extra Credit: Complete the Radian Unit Circle.

Label all ordered pairs formed by axes and multiples of 30 and 45 degrees, and all of these angles (in radians). Be creative and artistic; use color (for example pencils or markers) or cut images from magazines or the Web. This worksheet is posted online if you want to use the computer.