First Quarter Notes

Info Tech M$6 - 01

Stanley Switalski 9-6-06

Aim: How do we find trigonometric function values? Do Now: 2B + 4 = 4 – 2B +2B +2B 4B + 4 = 4 -4 -4 4B = 0 /4 /4 B=0 B 30 2 CM

2 CM H

A 60

D 2 CM

C 2 = B2 – A 2 4=H2–1 -1 - 1 3=H2 √3 = √H 2 √3 = H

C

1 CM 2 2 2 30 C = B2 – A 64 = B – 16 -16 - 16 48 = B 2 √48 = √B 2 4√3 = B

8

60 4

The ration of 30-60-90 is (30º) (60º) (90º) 1S √3S 2S

30 S√3

2S

60 S

So X - X√3 – 2X If X = 6 Then the ratio will be: 30

6 – 6 √3 – 2(6)

X

Y

60 6 (45º) (45º) (90º) 1 1 1√2 2 2 2√2 Ratio: X – X – X√2

45 1

45 1

45

9 – 9 – 9√2

C

D

9

Aim: What other trigonometric functions are there?

9-7-06

∆ ABC is a right triangle with measure of angle C = 90, AC = √3, BC = 1, and AB = 2 Write each trigonometric value as a fraction. Sin A = ½ SOHCAHTOA B Cos A= √3/2 Tan A= 1/√3 Sin B = √3/2 √3 Cos B= ½ 1 Tan B = √3/1 B C

A

2

Ø = theta (missing angle) Sin ø = cosecant (csc) = Hyp/opp Cos ø = secant (sec) = Hyp/adj Tan ø = contangent (cot) = Adj/opp

Ø C

4

A

Sin ø = 4/5 Csc ø = 5/4 Cos ø= 3/5 Sec ø = 5/3 Tan ø= 4/3 Cot ø = 3/4

5

Ø 3 Sin ø = 15/17 Csc ø = 17/15 Cos ø= 8/17 Sec ø = 17/8 Tan ø= 15/8 Cot ø = 8/15

Ø

8

17

15 (sin ø) · (tan ø) = (sin ø) / 1 · (sin ø) / (cos ø) = (sin 2 ø) / (cos ø)

Aim: How to find the values of reciprocal functions Reciprocal Trigonometric Functions CSC Ø = 1 / sin ø SEC Ø = 1 / cos ø *** TAN Ø = Sin ø / Cos ø

9/8/06 COT Ø = 1 / tan ø

Ø 30 60 90 To find these values sec ø of 30 = 1/ cos (30) Csc ø 2 1.2 1.4 Sec ø 1.2 2 1.4 Cot ø 1.7 .6 1 More Examples: 1.) sec 150° = 1 / cos (150) = -1.15 2.) cot 240° = 1 / tan (240) = .6 3.) cot 45 · csc 45 = 1 / tan (45) · 1 / sin (45) = 1.4 1 1.414 2 2 4.) cot 30° = 1/ tan (30) = 1 / (1.73)2 = 3 Write each expression in simplest form 1. Cot ø = Cos ø / sin ø 2. Cot ø · Sec ø = Cos ø / sin ø · 1 / cos ø = 1 / sin ø 3. Tan ø / Csc ø = Sin ø / Cos ø · Sin ø / 1 = Sin 2 ø / cos ø

Aim: How to use trigonometric identity to simplify expressions.

9/11/06

Do Now: Explain what it means to simplify a trigonometric expression. To put the trigonometric expression in the smallest possible radical form. Simplifying an expression that contains trigonometric functions means that the expression is written as a numerical value or in terms of a single trigonometric function if possible. Trigonometric identity – An equation that is true for all values of the variables.

Quotient Identities Reciprocal Identites

Basic Trigonometric Tan ø = sin ø / cos ø Cot ø = cos ø / sin ø CSC Ø = 1 / Sin ø Sec Ø = 1 / Cos ø Cot Ø = 1 / Tan ø

1.) (Csc ø) (Cos ø) (Tan ø) 1 / sin ø cos ø / 1 sin ø / cos ø = 1 2.) Cos ø · Csc ø cos ø / 1 1 / sin ø = cos ø / sin ø = cot ø 3.) Cot ø / Csc ø = (cos ø / sin ø) / 1 / sin ø Cos ø/ Sin ø · Sin ø / 1 = Cos ø 4.) Sec ø / Cot ø = 1 / (cos ø) / (cos ø / sin ø) 1 / (cos ø) · sin ø / cos ø = Sin ø / Cos 2 ø 1.) 2.) 3.) 4.) 5.)

Cot ø · Sec ø = (cos ø / sin ø) · 1 / (cos ø) = 1 / sin ø = csc ø Cot ø · Sin ø = (cos ø / sin ø) · sin ø / 1 = cos ø Cos ø / 1 · Sin ø / Cos ø = sin ø Sec ø · Csc ø = 1 / cos ø · 1 / sin ø Tan ø / Csc ø Sin ø / Cos ø · Sin ø / 1 = Sin 2 ø / Cos ø

Aim: Radian to angle.

9/22/06 510 – 360 = 150

510º

150/ 180 = X / π 5/6=X/π 5 π / 6 = 6X / 6 5 π/6 = X

Rewrite each in radians a) 240º 240/ 180 = X / π 4 / 3 = X / π 4 π = 3 X / 3 4 π / 3 = X b) 90º 90 / 180 = ½ ½ = x / π π = 2 X / 2 π / 2 = X c) – 90º = - π d) 135º = 135 / 180 = X / π 135 π = 180 X / 180 135 π / 180 = X To Change radians to degrees Radians · 180 / π Examples: 5 π / 8 · 180 / π = 5 / 2 · 45/ 1 = 225 / 2 = 112.5º 16 π / 5 = 180 / π = 16 · 36 = 576 º

3π/4=X