Section 1.2 Trigonometric Ratios
description
Transcript of Section 1.2 Trigonometric Ratios
![Page 1: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/1.jpg)
Section 1.2Trigonometric Ratios
Section 1.2Trigonometric Ratios
![Page 2: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/2.jpg)
Objectives:
1. To state and apply the Pythagorean theorem.
2. To define the six trigonometric ratios.
Objectives:
1. To state and apply the Pythagorean theorem.
2. To define the six trigonometric ratios.
![Page 3: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/3.jpg)
Pythagorean TheoremIn right ABC, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
a2 + b2 = c2
BB
AA
CC
![Page 4: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/4.jpg)
hypotenusehypotenuseleg opposite Aleg opposite A
sine of A =sine of A =
hypotenusehypotenuseleg adjacent Aleg adjacent A
cosine of A =cosine of A =
leg adjacent Aleg adjacent Aleg opposite Aleg opposite A
tangent of A =tangent of A =
Trigonometric Ratios
![Page 5: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/5.jpg)
SOHCAHTOA
SOHCAHTOA
ineppositeypotenuseosinedjacentypotenuseangentppositedjacent
![Page 6: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/6.jpg)
hhoo
AAsinsin ==hhaa
AAcoscos ==aaoo
AAtantan ==
A
opposite
adjacent
hypotenuse
Trigonometric RatiosTrigonometric Ratios
![Page 7: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/7.jpg)
cosAcosA11
secant of A = secA = secant of A = secA =
sinAsinA11
cosecant of A = cscA = cosecant of A = cscA =
tanAtanA11
cotangent of A = cotA = cotangent of A = cotA =
Reciprocal RatiosReciprocal Ratios
![Page 8: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/8.jpg)
oohh
AAcsccsc ==aahh
AAsecsec ==ooaa
AAcotcot ==
Reciprocal RatiosReciprocal Ratios
A
opposite
adjacent
hypotenuse
![Page 9: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/9.jpg)
EXAMPLE 1 Find the six trigonometric ratios for G in right EFG.
EXAMPLE 1 Find the six trigonometric ratios for G in right EFG.
6688
GG
EE FF
g2 + e2 = f2
g2 + 62 = 82
g2 + 36 = 64g2 = 28g = 2 7
![Page 10: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/10.jpg)
Practice Question: Find the six trigonometric ratios for E in right EFG.
Practice Question: Find the six trigonometric ratios for E in right EFG.
991111
GG
EE FF
sin E =
1. 2.
3. 4.
911
2 1011
9 1020
2 109
![Page 11: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/11.jpg)
yy
xx
rr
P(x,y)P(x,y)
![Page 12: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/12.jpg)
yyxx
oppoppadjadj
cotcot ====xxyy
adjadjoppopp
tantan ====
xxrr
adjadjhyphyp
secsec ====rrxx
hyphypadjadj
coscos ====
yyrr
oppopphyphyp
csccsc ====rryy
hyphypoppopp
sinsin ====
Trigonometric RatiosTrigonometric Ratios
![Page 13: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/13.jpg)
EXAMPLE 2 Find the six trigonometric ratios for a 90º angle.EXAMPLE 2 Find the six trigonometric ratios for a 90º angle.
PP90°90°
P = (0, 1)x=0, y=1, r=1cos = 0sin = 1tan = und.
![Page 14: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/14.jpg)
EXAMPLE 2 Find the six trigonometric ratios for a 90º angle.EXAMPLE 2 Find the six trigonometric ratios for a 90º angle.
PP90°90°
P = (0, 1)x=0, y=1, r=1sec = und.csc = 1cot = 0
![Page 15: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/15.jpg)
Practice Question: Find the six trigonometric ratios for a 180º angle.Practice Question: Find the six trigonometric ratios for a 180º angle.
PP
180°180°
sin = 1. -1 2. 03. 1 4. und.
![Page 16: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/16.jpg)
Special TrianglesSpecial Triangles
22cc ==
4545
11
45451
a2 + b2 = c2
12 + 12 = c2
1 + 1 = c2
c2 = 2
a2 + b2 = c2
12 + 12 = c2
1 + 1 = c2
c2 = 2
![Page 17: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/17.jpg)
2
1
Special TrianglesSpecial Triangles
22cc ==
4545
11
45451
sin 45° =2
2
![Page 18: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/18.jpg)
22
Special TrianglesSpecial Triangles
22cc ==
4545
11
45451
sin 45° =
cos 45° =
tan 45° =
22
1
![Page 19: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/19.jpg)
2
Special TrianglesSpecial Triangles
22cc ==
4545
11
45451
csc 45° =
sec 45° =
cot 45° =
2
1
![Page 20: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/20.jpg)
2222
sin 45° =sin 45° =
cos 45° =cos 45° =
tan 45° =tan 45° =
2222
11
22csc 45° =csc 45° =
sec 45° =sec 45° =
cot 45° =cot 45° =
22
11
![Page 21: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/21.jpg)
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
a2 + b2 = c2
12 + b2 = 22
1 + b2 = 4b2 = 3
a2 + b2 = c2
12 + b2 = 22
1 + b2 = 4b2 = 3
![Page 22: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/22.jpg)
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
tan 30° =3
1
3
3
![Page 23: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/23.jpg)
3333
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
tan 30° =
sin 30° =
cos 30° =2233
2211
![Page 24: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/24.jpg)
33
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
cot 30° =
csc 30° =
sec 30° =
22
333322
![Page 25: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/25.jpg)
3333
tan 30° =
sin 30° =
cos 30° =2233
2211
33cot 30° =
csc 30° =
sec 30° =
22
333322
![Page 26: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/26.jpg)
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
sin 60° =
cos 60° =
tan 60° =
2233
2211
33
![Page 27: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/27.jpg)
3333
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
csc 60° =
sec 60° =
cot 60° =
333322
22
![Page 28: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/28.jpg)
sin 60° =
cos 60° =
tan 60° =
2233
2211
333333
csc 60° =
sec 60° =
cot 60° =
333322
22
![Page 29: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/29.jpg)
Homework:
pp. 12-13
Homework:
pp. 12-13
![Page 30: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/30.jpg)
►A. Exercises1. Find the six trig. ratios for both
acute angles in each triangle.
►A. Exercises1. Find the six trig. ratios for both
acute angles in each triangle.
5
C
A
B
2
21
![Page 31: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/31.jpg)
12
LM
N2
1st-solve for side n1st-solve for side n22 + n2 = 12222 + n2 = 122n2 = 122-22 = 144-4 = 140n2 = 122-22 = 144-4 = 140
n
►A. Exercises3. Find the six trig. ratios for both
acute angles in each triangle.
►A. Exercises3. Find the six trig. ratios for both
acute angles in each triangle.
353522140140nn ====
![Page 32: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/32.jpg)
#3.#3.
12
LM
N2
352
![Page 33: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/33.jpg)
►A. Exercises7. Find the six trig. functions for an
angle in standard pos. whose terminal ray passes through the point (-6, -1).
►A. Exercises7. Find the six trig. functions for an
angle in standard pos. whose terminal ray passes through the point (-6, -1).
-6-6-1-1
3737
![Page 34: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/34.jpg)
-6-6-1-1
3737
sin = csc =
cos = sec =
tan = cot =
![Page 35: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/35.jpg)
►B. Exercises15. Find the six trig. ratios for the
quadrantal angle measuring 180°.
►B. Exercises15. Find the six trig. ratios for the
quadrantal angle measuring 180°.
![Page 36: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/36.jpg)
(-1, 0)(-1, 0)
►B. Exercises15. Find the six trig. ratios for the
quadrantal angle measuring 180°.
►B. Exercises15. Find the six trig. ratios for the
quadrantal angle measuring 180°.
![Page 37: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/37.jpg)
■ Cumulative Review30. Give the distance between (2, 7) and
(-3, -1).
■ Cumulative Review30. Give the distance between (2, 7) and
(-3, -1).
![Page 38: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/38.jpg)
■ Cumulative Review31. Give the midpoint of the segment
joining (2, 7) and (-3, -1).
■ Cumulative Review31. Give the midpoint of the segment
joining (2, 7) and (-3, -1).
![Page 39: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/39.jpg)
■ Cumulative Review32. Give the angle coterminal with 835
if 0 360.
■ Cumulative Review32. Give the angle coterminal with 835
if 0 360.
![Page 40: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/40.jpg)
■ Cumulative Review33. Convert 88 to radians.■ Cumulative Review33. Convert 88 to radians.
![Page 41: Section 1.2 Trigonometric Ratios](https://reader035.fdocuments.net/reader035/viewer/2022081504/56814edf550346895dbc7442/html5/thumbnails/41.jpg)
■ Cumulative Review34. If sec = 7, find cos .■ Cumulative Review34. If sec = 7, find cos .