Transcript Trigonometry - Miss Thornton`s Homework Web Page

Trigonometry
Definition:
• Trigonometry is the art of studying
triangles (in particular, but not limited to,
right triangles)
• Trigonometry makes use of both the
angles and the side lengths
• Deals with the relationships between the
angles and side lengths of a triangle
• The sine, cosine and tangent of acute
angles in a 90° triangle show how the side
lengths are related to the angles
• Before learning the key formulas in
trigonometry, it is of absolute importance
that some terms are understood
• Because we are dealing with right
triangles, you are already familiar with one
very important right triangle theorem:
– The Pythagorean Theorem
a² + b² = c²
• In every right triangle, because one of the angles
measures 90°, then logically the other two angles
must add up to 90°
A
B
C
Because m<B = 90° then m<A + m<C = 90° (since there are 180° in every triangle)
Hypotenuse:
– The side that is opposite the
right angle
– The longest side in the right
triangle
A
Opposite Side:
– The side that is opposite of a
given angle
– Ex: Side AB is opposite m<C
Side BC is opposite m<A
Adjacent Side:
– The side that is neither
hypoteneuse or opposite
Ex: Side BC is adjacent to m<C
Side AB is adjacent to m<A
B
C
Example:
Fill in the blanks in the following questions:
A
Hypotenuse: _________________
Opposite m<A: _________________
Adjacent m<A: _________________
B
Opposite m<C: __________________
Opposite m<C: __________________
C
These three definitions of the sides are of
utmost importance in
trigonometry
They are at the root of finding every angle in
a right triangle
Class work and homework:
Hand out 1 and Math 3000 page 182 # 1
Trigonometric Ratios in a
Right Triangle
Sine of an acute angle
• The sine of an acute angle is equal to the
ratio of the measure of the opposite side to
that angle over the measure of the
hypotenuse
• The sine of angle A is written sin A
Cosine of an Acute angle
• The cosine of an acute angle is equal to
the ratio of the measure of the adjacent
side to that angle over the measure of the
hypotenuse
• The cosine of angle A is written cos A
Tangent of an acute angle
• The tangent of an acute angle is equal to
the ratio of the measure of the opposite
side to that angle over the measure of the
adjacent side to that angle
• The tangent of angle A is written tan A
The MOST important Gibberish
word you will need to remember in
life
SOH – CAH - TOA
Opposite
sin A 
Hypotenuse
A
Adjacent
cos A 
Hypotenuse
Opposite
tan A 
Adjacent
B
C
Example:
o p p o site
1
sin 3 0 

h yp o ten u se 2
a d ja cen t
3
co s3 0 

h yp o ten u se
2
o p p o site
1
t an3 0 

a d ja cen t
3
3
sin 6 0 
2
1
co s 6 0 
2
3
t an 6 0 
1
A
60°
2
1
30°
B
C
3
• When writing a trigonometric ratio, we can
write the measure of the angle when it is
known.
• Thus:
• The sine of angle B measuring 30° is
written sin 30°
Homework
P. 182 # 1
P. 184 # 2
(2) Review
• Definitions:
• Trig Ratios :
SOH CAH TOA
• Hand out 1 – Identifying Opposite,
Adjacent, and Hypotenuse & Sine, Cosine,
Tangent Problems
• Hand out 2 – Find the Trig Ratio
3. Using your Calculator
Using Your Calculator
1. The keys sin, cos, tan on the calculator
enable you to calculate the value of sin
A, cos A, or tan A knowing the
measure of angle A
So if you know the measure of an angle
you can use the sin, cos, or tan
buttons on your calculator in order to
calculate its value
Formulas for 90° triangle
Formulas to find a missing side
Formulas to find a missing angle
(hyp)² = (opp)² + (adj)²
Sin( A) opp

1
hyp
Cos( A) adj

1
hyp
Tan( A) opp

1
adj
 opp 
A  sin 1 

 hyp 
 adj 
A  cos1 

 hyp 
 opp 
A  tan 1 

 adj 
Summary
• The three new formulas for 90° triangles
Sin, Cos, Tan are used to find a
missing side length in a right triangle
2. The keys sin-1, cos-1, tan-1 on the
calculator enable you to calculate the
measure of angle A knowing sin A
• So if know sin A, cos A, or tan A,
you can calculate the measure of
angle A
Summary
• If we take the inverse of each formula, we
can find the missing side angle in a
90° triangle
• The symbol for the inverse of
sin (A) is sin-1; cos (A) is cos-1;
tan (A) is tan-1
Example
sin 30º = 0.5 and sin-1 (0.5) = 30º
Class work
• Mathematics 3000 , page 185 numbers 4
and 5
• Hand out number 2
Finding Missing Angles using
Trigonometry Ratios
Formulas for 90° triangle
Formulas to find a missing side
Formulas to find a missing angle
(hyp)² = (opp)² + (adj)²
Sin( A) opp

1
hyp
Cos( A) adj

1
hyp
Tan( A) opp

1
adj
 opp 
A  sin 1 

 hyp 
 adj 
A  cos1 

 hyp 
 opp 
A  tan 1 

 adj 
In a Right Triangle
1. Find the acute angle A when its opposite
side and the hypotenuse are known
requires the use of sin A
SOH – Opposite/hypotenuse
sin A =
4
5
M<A=sin-1 ( )=53.1º
4
5
2. Finding the acute angle A when its
adjacent side and the hypotenuse are
known requires the use of cos A
Cos = adjacent/hypotenuse
Cos A=
3
4
m<A = cos-1 ( ) = 41.4º
3
4
3. Finding the acute angle A when its
opposite side and adjacent side are
known requires the use of tan A
tan = opposite/adjacent
Tan A =
3
2
m<A=tan-1 ( ) = 56.3º
3
2
How to:
1. Label known sides H,O,A
2. Select sine, cosine or tangent depending
on information known
3. Set up ratio – leave either as a ratio OR
reduce to decimal round to 4 places
(thousandth)
4. M<a = inverse of sine, cos or tan and
that ratio or decimal
5. Result is your missing angle
Class work
• Mathematics 3000 page 186, activity 4
• Mathematics 3000, page 187, number 8
• Handout number 4
Finding Missing Sides Using
Trigonometric Ratios
Formulas for 90° triangle
Formulas to find a missing side
Formulas to find a missing angle
(hyp)² = (opp)² + (adj)²
Sin( A) opp

1
hyp
Cos( A) adj

1
hyp
Tan( A) opp

1
adj
 opp 
A  sin 1 

 hyp 
 adj 
A  cos1 

 hyp 
 opp 
A  tan 1 

 adj 
In a right triangle
1. Finding the measure x of side BC
opposite to the known angle A, knowing
also the measure of the hypotenuse,
requires the use of sin A
Remember: SOH
*****Cross Multiply*****
Sin 50º=
x
5
x=5sin50º = 3.83 cm
Finding the measure y of side AC adjacent
to the known Angle A, knowing also the
measure of the hypotenuse, requires the
use of cos A
Remember: cos = adjacent/hypotenuse
*****Cross Multiply*****
Cos 50º =
y
5
y = 5 cos 50º = 3.21 cm
3. Finding the measure x of side BC
opposite to the known angle A, knowing
also the measure of the adjacent side to
angle A, requires the use of tan A
remember tan=opposite/adjacent
***cross multiply***
x
tan 30º =
4
x = 4 tan 30º = 2.31 cm
Class work
• Mathematics page 185, activity 3
• Mathematics page 186, numbers 6,7
• Handout number 6
Class Work and Homework
• Page 186, numbers 6 and 7
Solving a triangle
To determine the measure of all
its sides and angles
Class work and homework
• Math3000 page187, number 9
• Page 188, number 10, 11,12
Sine Law
• The sides in a triangle are directly
proportional to the sine of the opposite
angles to these sides
a
b
c


sin A sin B sin C
It is also true that:
sin A sin B sin C


a
b
c
• The sine law can be
used to find the
measure of a missing
side or angle
1st Case
• Finding a side when we know two angles
and a side
We calculate the measure x of AC
x
15
15sin 50


x 
 13.27cm
sin 50 sin 60
sin 60
How to:
1. Place Measurement x over sin known
angle
2. Equal to
3. Measurement known side over sin of
known angle
4. Cross multiply and divide to find unknown
measurement
5. Calculate.
2nd Case
• Finding an angle when we know two
sides and the opposite angle to one of
these two sides
• We calculate the measure of angle B
10
13
10sin 50

sin B 
 0.5893 mB  36
sin B sin 50
13
• Make sure you have opposite angles and
side measurements. Remember total
inside angles must equal 180º
How to calculate if need to find
an angle:
1. Place side measurement known over sin
of angle we wish to know
2. Equal to
3. side measurement over sin angle we
know
4. Cross multiply and divide to find x
5. To calculate angle –sin x = angle. Don’t
forget unit i.e.º
Class work and homework
1. Math 3000, page 190, number 1 a and b
– we will do altogether
2. Class work: page 190, number 1 c-f and
number 2
3. Finish all above work tonight
The sine of an obtuse angle
• The trigonometric
functions (sine,
cosine, etc.) are
defined in a right
triangle in terms of an
acute angle. What,
then, shall we mean
by the sine of an
obtuse angle ABC?
• The sine of an obtuse angle is
defined to be the sine of its
supplement.
• How to find the measure of the degree of
an obtuse angle:
• Follow the procedure you have learned
so far, then subtract that angle from
180º
10 cm
22º
18.6 cm
18.6
10

sin 22 sin w
18.6sin 22
10
 .697
  sin .697
 44.2
 180  44.2  135.5
in
• Class work – page 190 #4,6