trigonometric ratios in right triangles m. bruley
TRANSCRIPT
Trigonometric Ratios in Right Triangles
M. Bruley
Trigonometric Ratios are based on the Concept of Similar Triangles!
All 45º- 45º- 90º Triangles are Similar!
45 º
2
2
22
45 º
1
1
2
45 º
1
2
1
2
1
All 30º- 60º- 90º Triangles are Similar!
1
60º
30º
½
23
32
60º
30º
2
4
2
60º
30º
1
3
All 30º- 60º- 90º Triangles are Similar!
10 60º
30º
5
35
2 60º
30º1
3
160º
30º 21
23
The Tangent Ratio
c a
b
c’ a’
b’
If two triangles are similar, then it is also true that: '
'
b
a
b
a
The ratio is called the Tangent Ratio for angle b
a
Naming Sides of Right Triangles
The Tangent Ratio
Tangent Tangent Adjacent
Opposite
There are a total of six ratios that can be madeThere are a total of six ratios that can be madewith the three sides. Each has a specific name.with the three sides. Each has a specific name.
The Six Trigonometric Ratios(The SOHCAHTOA model)
Adjacent
OppositeTangentθ
Hypotenuse
AdjacentCosineθ
Hypotenuse
OppositeSineθ
The Six Trigonometric Ratios
Adjacent
OppositeTangentθ
Hypotenuse
AdjacentCosineθ
Hypotenuse
OppositeSineθ
Opposite
AdjacentCotangentθ
Adjacent
HypotenuseSecantθ
Opposite
HypotenuseCosecantθ
The Cosecant, Secant, and Cotangent of The Cosecant, Secant, and Cotangent of are the Reciprocals of are the Reciprocals of
the Sine, Cosine,and Tangent of the Sine, Cosine,and Tangent of
Solving a Problem withthe Tangent Ratio
60º
53 ft
h = ?
We know the angle and the We know the angle and the side adjacent to 60º. We want to side adjacent to 60º. We want to know the opposite side. Use theknow the opposite side. Use thetangent ratio:tangent ratio:
ft 92353
531
3
5360tan
h
h
h
adj
opp
1
2 3
Why?
Trigonometric Functions on a Rectangular Coordinate System
x
y
Pick a point on theterminal ray and drop a perpendicular to the x-axis.
(The Rectangular Coordinate Model)(The Rectangular Coordinate Model)
Trigonometric Functions on a Rectangular Coordinate System
x
y
Pick a point on theterminal ray and drop a perpendicular to the x-axis.
ry
x
The adjacent side is xThe opposite side is yThe hypotenuse is labeled rThis is called a REFERENCE TRIANGLE.
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
Trigonometric Values for angles in Quadrants II, III and IV
x
yPick a point on theterminal ray and drop a perpendicular to the x-axis.
y
x
r
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
Trigonometric Values for angles in Quadrants II, III and IV
x
yPick a point on theterminal ray and raise a perpendicular to the x-axis.
Trigonometric Values for angles in Quadrants II, III and IV
x
yPick a point on theterminal ray and raise a perpendicular to the x-axis.
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
x
ry
Important! The is
ALWAYS drawn to the x-axis
Signs of Trigonometric Functions
x
y
AAll are positive in QI
TTan (& cot) are positive in QIII
SSin (& csc) are positive in QII
CCos (& sec) are positive in QIV
Signs of Trigonometric Functions
x
y
AAll
TTake
SStudents
CCalculus
is a good way toremember!
Trigonometric Values for Quadrantal Angles (0º, 90º, 180º and 270º)
x
y
º
Pick a point one unit from the Origin.
(0, 1)
r
x = 0y = 1r = 1
090cotundefined is 90tan
undefined is 90sec090cos
190csc190sin
Trigonometric Ratios may be found by:
45 º
1
1
2Using ratios of special trianglesUsing ratios of special triangles
145cot145tan
245sec2
145cos
245csc2
145sin
For angles other than 45º, 30º, 60º or Quadrantal angles, you will For angles other than 45º, 30º, 60º or Quadrantal angles, you will need to use a calculator. (Set it in Degree Mode for now.)need to use a calculator. (Set it in Degree Mode for now.)
For Reciprocal Ratios, use the facts:For Reciprocal Ratios, use the facts:
tan1cot
cos1sec
sin1csc
Acknowledgements
This presentation was made possible by training and equipment provided by an Access to Technology grant from Merced College.
Thank you to Marguerite Smith for the model. Textbooks consulted were:
Trigonometry Fourth Edition by Larson & Hostetler Analytic Trigonometry with Applications Seventh
Edition by Barnett, Ziegler & Byleen