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Section 1.2 Trigonometric Ratios. Objectives: 1.To state and apply the Pythagorean theorem. 2.To define the six trigonometric ratios. A. C. B. Pythagorean Theorem - PowerPoint PPT Presentation
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Section 1.2Trigonometric Ratios
Section 1.2Trigonometric Ratios
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.
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
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
SOHCAHTOA
SOHCAHTOA
ineppositeypotenuseosinedjacentypotenuseangentppositedjacent
hhoo
AAsinsin ==hhaa
AAcoscos ==aaoo
AAtantan ==
A
opposite
adjacent
hypotenuse
Trigonometric RatiosTrigonometric Ratios
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
oohh
AAcsccsc ==aahh
AAsecsec ==ooaa
AAcotcot ==
Reciprocal RatiosReciprocal Ratios
A
opposite
adjacent
hypotenuse
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
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
yy
xx
rr
P(x,y)P(x,y)
yyxx
oppoppadjadj
cotcot ====xxyy
adjadjoppopp
tantan ====
xxrr
adjadjhyphyp
secsec ====rrxx
hyphypadjadj
coscos ====
yyrr
oppopphyphyp
csccsc ====rryy
hyphypoppopp
sinsin ====
Trigonometric RatiosTrigonometric Ratios
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.
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
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.
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
2
1
Special TrianglesSpecial Triangles
22cc ==
4545
11
45451
sin 45° =2
2
22
Special TrianglesSpecial Triangles
22cc ==
4545
11
45451
sin 45° =
cos 45° =
tan 45° =
22
1
2
Special TrianglesSpecial Triangles
22cc ==
4545
11
45451
csc 45° =
sec 45° =
cot 45° =
2
1
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
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
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
tan 30° =3
1
3
3
3333
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
tan 30° =
sin 30° =
cos 30° =2233
2211
33
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
cot 30° =
csc 30° =
sec 30° =
22
333322
3333
tan 30° =
sin 30° =
cos 30° =2233
2211
33cot 30° =
csc 30° =
sec 30° =
22
333322
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
sin 60° =
cos 60° =
tan 60° =
2233
2211
33
3333
33bb ==
Special TrianglesSpecial Triangles
6060
22
11
3030
csc 60° =
sec 60° =
cot 60° =
333322
22
sin 60° =
cos 60° =
tan 60° =
2233
2211
333333
csc 60° =
sec 60° =
cot 60° =
333322
22
Homework:
pp. 12-13
Homework:
pp. 12-13
►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
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 ====
#3.#3.
12
LM
N2
352
►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
-6-6-1-1
3737
sin = csc =
cos = sec =
tan = cot =
►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°.
(-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°.
■ Cumulative Review30. Give the distance between (2, 7) and
(-3, -1).
■ Cumulative Review30. Give the distance between (2, 7) and
(-3, -1).
■ 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).
■ Cumulative Review32. Give the angle coterminal with 835
if 0 360.
■ Cumulative Review32. Give the angle coterminal with 835
if 0 360.
■ Cumulative Review33. Convert 88 to radians.■ Cumulative Review33. Convert 88 to radians.
■ Cumulative Review34. If sec = 7, find cos .■ Cumulative Review34. If sec = 7, find cos .