aim: the six trigonometric functions course: alg. 2 & trig. aim: what does sohcahtoa have to do...
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Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Aim: What does SOHCAHTOA have to do with our study of right triangles?
Do Now:
Key terms: adjacent, opposite & hypotenuse
4
3
5
C B
AWhat are the following ratios?
ACAB =
ACCB =
BCAB =
3545
43
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Trigonometry Basics - Sine
In a right ABC with right angle BCA
• The sine of angle B, written sine B, is defined as
hypotenusetheoflength
Boppositelegtheoflength
BA
ACB sin
4
3
5
C B
Asin B =
ACAB
=54
3sin
5
BCA
AB
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Sine’s Reciprocal
What is the reciprocal of sin?What is the reciprocal of 3? 1/3
1/sin
the reciprocal of sin has a special name:
sin =
2
2csc = ?
2
2=
2 2
2
2ex.
using the calculator to find csc 53º:find csc 53º:
sin x -153 ENTER ENTER
Display: 1.252135658
Method 1
cosecant
NOTE: csc sin 1
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Trigonometry Basics - Cosecant
In a right ABC with right angle BCA
• The cosecant of angle B, written csc B, is defined as
cscBA length of the hypotenuse
BAC length of the leg opposite B
4
3
5
C B
Acsc B =
ABAC
=45
5csc
3
ABA
BC
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Trigonometry Basics - Cosine
In a right ABC with right angle BCA
• The cosine of angle B, written cos B, is defined as
hypotenusetheoflength
Btoadjacentlegtheoflength
BA
BCB cos
4
3
5
C B
Acos B =
BCAB
=53
4cos
5
ABA
AC
Recall: 3
sin5
BCA
AB
the sine of an acute angle has the same value as the cosine of its complement.
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Cosine’s Reciprocal
The reciprocal of cosine is the secant :
sec 1
cos
sec = ? ex. cos =
1
22
= ? 2nd ÷cos -1 ENTER1 2
Display: 60
using the calculator to find sec :Find sec (-38º):
( – )1 cos ENTER
Display: 1.269018215
Method 2
÷ 38
NOTE: sec cos 1
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Trigonometry Basics - Secant
In a right ABC with right angle BCA
• The secant of angle B, written sec B, is defined as
4
3
5
C B
Asec B =
ABBC
=35
sec
AB length of the hypotenuseB
BC length of the leg adjacent B
5sec
4
ABA
AC
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Trigonometry Basics - Tangent
• The tangent of angle B, written tan B, is defined as
Btoadjacentlegtheoflength
Boppositelegtheoflength
BC
ACB tan
4
3
5
C B
A
In a right ABC with right angle BCA
tan B =ACBC
=34
3tan
4
BCA
AC
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Tangent’s Reciprocal
The reciprocal of tan is the cotangent :
cot 1
tan
cot = ? ex. tan =
3
3
3
3
3=
= ? 2nd tan -1 ENTER Display: 60
3
Using the calculator to find cot :Find cot 257º:
tan x -1257 ENTER ENTER
Display: .2308681911
Method 1
1 tan ENTER
Display: .2308681911
÷ 257Method 2
NOTE: cot tan 1
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Trigonometry Basics - Cotangent
• The cotangent of angle B, written cot B, is defined as
cotBC length of the leg adjacent to B
BAC length of the leg opposite B
4
3
5
C B
A
In a right ABC with right angle BCA
cot B =BCAC
=43
4cot
3
ACA
BC
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Meet Chief
Sine - SOH =Opposite
Hypotenuse
Tangent - TOA =Opposite
Adjacent
Cosine - CAH =Adjacent
Hypotenuse
SOH CAH TOA
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Trig. Relationships
Recall:
3sin
5
BCA
AB
the sine of an acute angle has the same value as the cosine of its complement.
4
3
5
C B
A
3cos
5
ABB
AC
sin A = cos B and cos A = sin B
The tangent of an acute angle is the reciprocal of the tangent of its complement
tan A · tan B = 1
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
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
Model Problem
10
100
6436
86
2
2
222
222
AB
AB
AB
AB
ABACBC
222 bac
TheoremnPythagorea
In right triangle ABC with right angle at C, BC = 6, and AC = 8. Find the three trigonometric functions of B.
8
6B C
A
leg adjacent to B
hypotenuse
leg opposite B
leg adjacent to B
10
leg opposite B
hypotenusesin B =
cos B =
tan B =
8
106
108
6
Aim: The Six Trigonometric Functions Course: Alg. 2 & Trig.
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’ 8’5o
x3’
o 3tan5
x
o
3
tan5x 34.29' 34 feet
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
2. If cot x = tan(x + 20o), find x.When the cotangent and tangent functions are equal in value, the angles must be complementary.
The sine of an angle has the same value as the cosine of its complement.
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
60 seconds in 1 minute
17o 43’05”
17 degrees 43 minutes 5 seconds
1 minute is 1/60th of a degree
1 second is 1/60th of a minute
o1 4317 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
3cos
2 What is measure of ?
2nd cos -1 ENTER3 22nd ÷
Calculator’s MODE must be in degrees
30o
sin 0.2478
cos 0.2249
tan 0.3987
What is measure of ?
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 mKTW = 90.
5
6
W
T K
What is the measure of K, to the nearest minute?
1) 33o33’ 2) 33o55’
3) 33o34’ 4) 33o56’