Unit 6 Lesson 6 Trigonometric Ratios CCSS G-SRT 6: Understand that by similarity, side ratios in...
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Unit 6 Lesson 6 Trigonometric Ratios CCSS G-SRT 6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT 7: Explain and use the relationship between the sine and cosine of complementary angles. G-SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Lesson Goals • Understand the three basic trigonometric ratios are based on the side lengths of a right triangle. • Write the three basic trigonometric ratios in fractional and decimal form. • Use a chart or calculator to find the three basic trig ratios for a given ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers
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Transcript of Unit 6 Lesson 6 Trigonometric Ratios CCSS G-SRT 6: Understand that by similarity, side ratios in...
Unit 6 Lesson 6 Trigonometric Ratios
CCSSG-SRT 6: Understand that by
similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT 7: Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Lesson Goals• Understand the three basic
trigonometric ratios are based on the side lengths of a right triangle.
• Write the three basic trigonometric ratios in fractional and decimal form.
• Use a chart or calculator to find the three basic trig ratios for a given angle.
ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers
PREV
IOU
SLY
IN
MAT
H• Ratios compare two quantities
by division
• In a right triangle, the hypotenuse is
the side opposite the right angle.
• The word “adjacent” means “next to”
hypotenuse
3 feet0.75
4 feet
Room 12 is adjacent to Room 11
Definition
Trigonometric RatioA ratio formed by comparing the lengths of two sides of a right triangle from the “viewpoint” of a given acute angle. Sine: leg length opposite an angle to the hypotenuse Cosine: leg length adjacent an angle to the hypotenuse Tangent: leg length opposite an angle to the leg
length adjacent the angle
Which leg is opposite ?A
Which leg is opposite ?B
You Try
B
CA
BC
AC
Which leg is adjacent ?B
You Try
AC
BC
Which leg is adjacent ?AB
CA
Definition
B
CA
tan A
cos A
sin A
BC
ACad
op
ja
posi
cent
te leg
leg
AC
ABh
a
y
dj
po
acen
ten
t leg
use
BC
ABh
o
y
pp
po
osit
ten
e leg
use
Definition
B
CA
tan B
cosB
sin B
AC
BCad
op
ja
posi
cent
te leg
leg
BC
ABh
a
y
dj
po
acen
ten
t leg
use
AC
ABh
o
y
pp
po
osit
ten
e leg
use
Trigonometric RatioThe Gabrielino–Tongva Indian Tribe that at one time lived in the
area which we now know as Fullerton used the word
SOHCAHTOA to help them remember the trigonometric ratios.
Sine
Opposite
Hypotenuse
Cosine
Adjacent
Hypotenuse
Tangent
Opposite
Adjacent
Bird DancersGabrielino/Tongva Indians of So Cal
Chris Markstrum
Fullerton Union High School mascot is "The Indians" and we often go by the nickname "The Tribe." You may modify this idea or go with whatever you use to help students remember the ratios.
BC
AC
5
12tan A
opposite leg
adjacent leg
cos A AC
AB
adjacent leg
hypotenuse 12
13
sin A BC
AB
opposite leg
hypotenuse
Example
5
13
Find the sin, cos, and tan for .A
0.3846
0.9231
0.4167
B
C A12
135
1
2
2
2
2
BC
AB
1
2
AC
AB
1
2
AC
BC
1
1tan B
opposite leg
adjacent leg
cosB adjacent leg
hypotenuse
sin B opposite leg
hypotenuse
ExampleFind the sin, cos, and tan for .B
B
C A1
12
CB
AB
7
25sin A
opposite leg
hypotenuse
You TryLeave answer as a fractionFind sin . .A
You TryLeave answer as a fractionFind cos . .E
2
2
RE
JEcosE
adjacent leg
hypotenuse
6
6 2
You TryLeave answer as a fractionFind tan . .T
MA
MT
8
12tanT
opposite leg
adjacent leg 2
3
Definition
Trigonometric IdentitiesAn equation involving trig ratios that is true for all acute angles.
2 2(sin ) (cos ) 1
sintan
cos
q is the “x” for angles
Example
sin A4
5
3
5cos A
2sin A 16
25
2cos A 9
25
2 2sin cosA A 16 9
25
25
25 1
A
BC 4
35
Proof2 2Prove: (sin ) (cos ) 1A A
statement reason
1. sin , cosa b
A Ac c
2 2
2 2
2 22. sin , cos
a bA A
c c
2 2
2 2
23. sin cos
a bA A
c
2 2 24. c a b
2
2 2
25. sin cos
cA A
c
2 26. sin cos 1A A
1. def Trig Ratios
2. Mult. Prop of =
3. Add. Prop of =
4. Pythagorean Th.
5. Substitution
6. Inverse Prop.
A
BC a
bc
Trigonometric Ratios
• Every acute angle has a sine, cosine, and tangent.• These ratios are usually written as a four-digit decimal approximation.• These ratios can be looked up in trigonometric tables or found with a scientific calculator.
Page 845also the last page of your notes.
Example
sin 34o = 0.5592
Example
tan 65o = 2.1445
Example
Using a scientific calculator.
Summary
What is meant by the sine, cosine, and tangent of an angle?
Today’s Assignment
p. 562: 10 – 20 e, 47, 48
Use a Trig Ratio Table or a calculator to find each value.
Use your knowledge of a
30-60-90 right triangle.
