Download Aim:

Aim: What does SOHCAHTOA have to do
with our study of right triangles?
Do Now:
A
4
C
3
What are the following ratios?
BC
3
AB = 5
5
AC
4
AB = 5
AC
4
B CB = 3
Key terms: adjacent, opposite & hypotenuse
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Trigonometry Basics - Sine
In a right DABC with right angle BCA
• The sine of angle B, written sine B,
is defined as
AC length of the leg opposite B
sin B 

BA
length of the hypotenuse
A
5
4
C
3
AC
4
sin B =
=
AB
5
BC 3
sin A 

AB 5
B
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Sine’s Reciprocal
What is the reciprocal of sin? 1/sin 
What is the reciprocal of 3? cosecant
1/3
the reciprocal of sin has a special name:
1
1
hypot .
csc 


sin oppos. oppos.
NOTE:  csc  sin   1 hypot .
2
ex. sin  =
2
2
2 2
=
csc  = ?
 2
2
2
using the calculator to find csc 53º:
Method 1
find csc 53º:
sin 53 ENTER x -1 ENTER
Display: 1.252135658
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Trigonometry Basics - Cosecant
In a right DABC with right angle BCA
• The cosecant of angle B, written csc B,
is defined as
BA
length of the hypotenuse
csc B 

AC length of the leg opposite B
A
5
4
C
3
AB
5
csc B =
=
AC
4
AB 5
csc A 

BC 3
B
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Trigonometry Basics - Cosine
In a right DABC with right angle  BCA
• The cosine of angle B, written cos B,
is defined as
BC length of the leg adjacent to B
cos B 

BA
length of the hypotenuse
A
BC
3
cos B =
=
AB
5
5
4
C
3
AB 4
cos A 

AC 5
B
the sine of an acute angle
BC 3
has the same value as the
A

Recall: sinAim:
The Six Trigonometric Functions
Alg. 2 & Trig.
AB 5 cosine ofCourse:
its complement.
Cosine’s Reciprocal
The reciprocal of cosine  is the secant :
1
NOTE:  sec  cos   1
sec 
cos 
1
ex. cos  =
2
=?
2nd
sec  = 2?
cos -1 1 ÷
2 ENTER
Display: 60
using the calculator to find sec  :
Method 2
Find sec (-38º):
1 ÷
cos ( – ) 38 ENTER
Display: 1.269018215
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Trigonometry Basics - Secant
In a right DABC with right angle  BCA
• The secant of angle B, written sec B,
is defined as
AB
length of the hypotenuse
sec B 

BC length of the leg adjacent B
A
AB
5
sec B =
=
BC
3
5
4
C
3
AB 5
sec A 

AC 4
B
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Trigonometry Basics - Tangent
In a right DABC with right angle  BCA
• The tangent of angle B, written tan B,
is defined as
AC
length of the leg opposite B
tan B 

BC length of the leg adjacent to B
A
5
4
C
3
AC
4
tan B =
=
BC
3
BC 3
tan A 

AC 4
B
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Tangent’s Reciprocal
The reciprocal of tan  is the cotangent :
1
cot  
tan
NOTE:  cot   tan   1
3
ex. tan  =
3
=?
tan -1
2nd
3
=
cot  = ?
3
3 ENTER
3
Display: 60
Using the calculator to find cot  :
Find cot 257º:
Method 1
tan
257 ENTER x -1 ENTER
Display: .2308681911
Method 2 1
÷
tan
257 ENTER
Display: .2308681911
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Trigonometry Basics - Cotangent
In a right DABC with right angle  BCA
• The cotangent of angle B, written cot B,
is defined as
BC length of the leg adjacent to B
cot B 

AC
length of the leg opposite B
A
5
4
C
3
BC
3
cot B =
=
AC
4
AC 4
cot A 

BC 3
B
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Meet Chief
SOH CAH TOA
Sine - SOH =
Cosine - CAH =
Opposite
Hypotenuse
Adjacent
Hypotenuse
Tangent - TOA =
Aim: The Six Trigonometric Functions
Opposite
Adjacent
Course: Alg. 2 & Trig.
A
4
Trig. Relationships
5
Recall:
BC 3
sin A 

AB 5
AB 3
cos B 

B
C
3
AC 5
the sine of an acute angle has the same value
as the cosine of its complement.
sin A = cos B and cos A = sin B
the tangent of an acute angle has the same
value as the cotangent of its complement.
tan A = cot B and cot A = tan B
The tangent of an acute angle is the reciprocal
of the tangent of its complement
tan A · tan B = 1 Course: Alg. 2 & Trig.
Aim: The Six Trigonometric Functions
Model Problem
In right triangle ABC with right angle at C,
BC = 6, and AC = 8. Find the three
trigonometric functions of  B.
A
Pythagorean Theorem
c 2  a 2  b2
BC  AC  AB
2
2
10
2
8
6 2  8 2  AB 2
36  64  AB 2
B
100  AB 2
AB  10
sin B =
cos B =
tan B =
Aim: The Six Trigonometric Functions
6
C
leg opposite B 8
hypotenuse 10
leg adjacent to B 6
hypotenuse
10
leg opposite B 8
Course: Alg. 2 & Trig.
leg adjacent
to B 6
Model Problem
Park planners would like to build a bridge
across a creek. Surveyors have determined
that from 5 ft. above the ground the angle of
elevation to the top of an 8ft. pole on the
opposite side of the creek is 5o. Find the
length of the bridge to the nearest foot.
5’
x
5o
3’
8’
3
tan5 
x
3
 34.29' 34 feet
x
o
tan5
o
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Model Problems
1. sin 24o is equivalent to
a) cos 24o b) sin 66o c) cos 660 d) 1/sin 240
The sine of an angle has the same value as
the cosine of its complement.
2. If cot x = tan(x + 20o), find x.
When the cotangent and tangent
functions are equal in value, the angles
must be complementary.
x + (x + 20) = 90
2x + 20 = 90
x = 70
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Degrees, Minutes & Seconds
3600 in a circle
60 minutes in 1 degree
1 minute is 1/60th of a degree
60 seconds in 1 minute
1 second is 1/60th of a minute
17o 43’05”
17 degrees 43 minutes 5 seconds
o
43
 1 
17 43'  17  43 
 17
 17.716

60
 60 
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Model Problem
Find cos 17o 43’ to 4 decimal places
Find sin 20.30o to 4 decimal places
Find sin 20o 30’ to 4 decimal places
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Find An Angle Given a Trig Function Value
3
What is measure of ?
cos 
2
Calculator’s MODE must be in degrees
2nd
cos -1
2nd
3 ÷ 2 ENTER
30o
What is measure of ?
sin  0.2478
cos  0.2249
tan  0.3987
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Regents Prep
In triangle ABC, side a = 7, b = 6, and c = 8.
Find m B to the nearest degree.
1) 43o
2) 47o
3) 65o
4) 137o
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.
Regents Prep
In the diagram below of right triangle
KTW, KW = 6, KT = 5, and mKTW = 90.
W
6
T
5
K
What is the measure of K, to the
nearest minute?
1) 33o33’
2) 33o55’
3) 33o34’
4) 33o56’
Aim: The Six Trigonometric Functions
Course: Alg. 2 & Trig.