Transcript Right-Triangle-Trigonometry-with

Right Triangle Trigonometry
Section 4.3
Objectives
• Calculate any trigonometric function for
an angle in a right triangle given two
sides of the triangle.
• Calculate the length of the sides of a
right triangle given the measure of an
angle of a triangle.
• Solve word problems requiring right
triangles and trigonometric functions.
Vocabulary
• angle of elevation
the angle that an observer would raise his or her line of
sight above a horizontal line in order to see an object.
angle of elevation
• angle of depression
If an observer were UP ABOVE and needed to look down,
the angle of depression would be the angle that the person
would need to lower his or her line of sight.
angle of depression
Trigonometric Functions
The side of the triangle opposite
the right angle is always named
the hypotenuse.
If we label one of the non-right
angles x, then we can name the
legs of the triangle.
The leg that makes up part of the
angle is called the adjacent side.
The leg that is not one of the
sides of the angle is called the
opposite side.
opposite (for angle x)
Each of the sides of a right
triangle can be labeled relative to
one of the non-right angles of the
triangle.
hypotenuse
x
adjacent (for angle x)
Trigonometric Functions
If we use the other nonright angle in the triangle
(angle y), then the sides
that are opposite and
adjacent change.
The side that is the
hypotenuse never changes.
It is always opposite the
right angle.
adjacent (for angle y)
The names of the sides will
change if we change the
angle.
y
hypotenuse
opposite (for angle y)
Trigonometric Functions
opposite (for angle x)
Once we have named
our sides, we can
define each of the
trigonometric
functions as ratios
of the sides of the
triangle.
hypotenuse
x
adjacent (for angle x)
Trigonometric Functions
opposite
hypotenuse
• cos(x)

adjacent
hypotenuse
• tan(x)

opposite
adjacent
opposite (for angle x)
• sin(x)

hypotenuse
x
adjacent (for angle x)
Trigonometric Functions
hypotenuse
opposite
• sec(x)

hypotenuse
adjacent
• cot(x)

adjacent
opposite
opposite (for angle x)
• csc(x)

hypotenuse
x
adjacent (for angle x)
For the triangle
y
• Find sin(x)
To find the sin(x), we will
need to length of the side
opposite the angle x and the
length of the hypotenuse.
We have the length of the
opposite side, 3. To find the
length of the hypotenuse, we
can use the Pythagorean
Theorem.
3
c
x
5
32  52  c 2
9  25  c 2
34  c 2
 34  c
length is positive
34  c
continued on next slide
For the triangle
y
Now that we have the length of the
hypotenuse, we can answer all of the
questions.
• Find sin(x)
opposite
sin( x ) 
hypotenuse
3
sin( x ) 
34
34
3
x
5
continued on next slide
For the triangle
Notice that for this question the angle has
changed to angle y. This will change what is
opposite and what is adjacent.
y
3
34
• Find cos(y)
adjacent
cos( y ) 
hypotenuse
x
5
3
cos( y ) 
34
One thing to note here is that angle y is equal to 90 - x. It is also the
case that x is equal to 90 – y. the In general when this happens, we have
the following co-function identities:
sin( x )  cos( 90  x )
cos( x )  sin( 90  x )
or

sin( x )  cos   x 
2

and
or

cos( x )  sin   x 
2

continued on next slide
For the triangle
y
Now continuing with the tangent and
cotangent functions.
34
• Find tan(x)
opposite
tan( x ) 
adjacent
3
tan( x ) 
5
3
x
5
continued on next slide
For the triangle
y
Now continuing with the tangent and
cotangent functions.
• Find cot(y)
adjacent
cot( y ) 
opposite
3
34
x
3
cot( y ) 
5
5
One thing to note here is that angle y is equal to 90 - x. It is also the
case that x is equal to 90 – y. the In general when this happens, we have
the following co-function identities:
tan( x )  cot( 90  x )
cot( x )  tan( 90  x )
or

tan( x )  cot   x 
2

and
or

cot( x )  tan   x 
2

continued on next slide
For the triangle
Now continuing with the secant and cosecant
functions.
• Find sec(x)
hypotenuse
sec( x) 
adjacent
34
sec( x) 
5
y
34
3
x
5
continued on next slide
For the triangle
Now continuing with the secant and cosecant
functions.
• Find sec(x)
hypotenuse
csc( y ) 
opposite
y
3
34
x
5
34
csc( y ) 
5
One thing to note here is that angle y is equal to 90 - x. It is also the
case that x is equal to 90 – y. the In general when this happens, we have
the following co-function identities:
sec( x)  csc(90  x)
csc( x)  sec(90  x)
or


sec( x)  csc  x 
2

and
or


csc( x)  sec  x 
2

Identities
• Cofunction Identities
•sin(x) = cos(90-x)
•cos(x) = sin(90-x)
•tan(x) = cot(90-x)
•cot(x) = tan(90-x)
•sec(x) = csc(90-x)
•csc(x) = sec(90-x)
For the triangle below, if BC = 7
and the angle β = 60, find all the
missing angles and sides.
C
Let’s start by putting the information
that we know into the triangle picture.
β
60°
7
α
A
B
continued on next slide
For the triangle below, if BC = 7
and the angle β = 60, find all the
missing angles and sides.
C
Our next step will be to do the
easiest work. I think that the
easiest thing to find next is
the measure of the angle α.
Since the measure of the
angles of a triangle add up to
180, we can calculate α as
β
60°
7
α=180°-60°-90°=30°
We can now put this into the
triangle.
α 30°
A
B
continued on next slide
For the triangle below, if BC = 7
and the angle β = 60, find all the
missing angles and sides.
Now we need to find lengths of
sides of the triangle. It does not
matter which side we decide to
find first. Let’s find side AB. To
do this we need to use one of the
non-right angles. Either angle will
work. I am going to choose the
60° angle since that was the one
that was given. Side AB is
opposite the 60 ° angle. The
other side that we know is BC
A
which is adjacent to the 60 °
angle. We need the trigonometric
function which has both opposite
and adjacent in it. This function
is the tangent function.
C
β
60°
7
α 30°
B
continued on next slide
For the triangle below, if BC = 7
and the angle β = 60, find all the
missing angles and sides.
We will use the tan(60) to find
side AB.
C
β
60°
opposite

tan( 60 ) 
adjacent
opposite
tan( 60 ) 
7
7 tan( 60 )  opposite
7

12.12435565  opposite
α 30°
A
12.12435565
B
continued on next slide
For the triangle below, if BC = 7
and the angle β = 60, find all the
missing angles and sides.
Now we are ready to find side AC.
No matter which angle we use,
the side AC is the hypotenuse of
the triangle. You may be tempted
to use the Pythagorean Theorem
here, but I do not recommend it.
If you have made an error or just
rounded the length of the side
that was just found, the
Pythagorean Theorem will produce
an incorrect answer. Since we A
want to use numbers that were
not calculated, but instead given
in the problem, we will use the 60°
and the side BC = 7.
C
β
60°
7
α 30°
12.12435565
B
continued on next slide
For the triangle below, if BC = 7
and the angle β = 60, find all the
missing angles and sides.
The side BC = 7 is adjacent to the
60° angle. The side AC is the
hypotenuse. We need a
trigonometric function which has
both the hypotenuse and the
adjacent sides. This would be the
cosine function.
cos( 60 ) 
adjacent
hypotenuse
cos( 60 ) 
7
hypotenuse
(hypotenuse ) cos( 60 )  7
7
cos( 60 )
hypotenuse  14
hypotenuse 
C
β
60°
14
α 30°
A
12.12435565
7
B
A plane if flying at an elevation
of 27000 feet. It is within sight
of the airport and the pilot finds
that the angle of depression to
the airport is 25 degrees. Find
the distance between the plane
and the airport.
A plane if flying at an elevation of 27000 feet. It is
within sight of the airport and the pilot finds that the
angle of depression to the airport is 25 degrees. Find
the distance between the plane and the airport.
For this problem we should draw a picture and label what we know.
angle of elevation = 25°
27000 feet
distance from plane to
airport
airport
distance from a point on the
ground directly below the
plane to the airport
A plane if flying at an elevation of 27000 feet. It is
within sight of the airport and the pilot finds that the
angle of depression to the airport is 25 degrees. Find
the distance between the plane and the airport.
In order to use trigonometric functions, we need a non-right angle
inside the triangle. We know that angle and A and the angle of
elevation add together to be 90°. Thus we can calculate angle A
as A = 90° - 25° = 65°.
angle of elevation = 25°
A
27000 feet
distance from plane to
airport
airport
distance from a point on the
ground directly below the
plane to the airport
continued on next slide
A plane if flying at an elevation of 27000 feet. It is
within sight of the airport and the pilot finds that the
angle of depression to the airport is 25 degrees. Find
the distance between the plane and the airport.
Now we can answer the question. For the 65° angle, the side of
length 27000 feet is the adjacent side. The side marked as the
distance from the plane to the airport is the hypotenuse. A
trigonometric function that has both adjacent and hypotenuse is
the cosine function.
adjacent
cos( 65) 
hypotenuse
cos( 65) 
27000
hypotenuse
hypotenuse cos( 65)  27000
27000
cos( 65)
hypotenuse  63887.44275
hypotenuse 
angle of elevation = 25°
65°
distance from plane to
airport
27000 feet
airport
distance from a point
on the ground directly
below the plane to the
airport
The distance from the plane to the
airport is 63887.44275 feet
continued on next slide
A plane if flying at an elevation of 27000 feet. It is
within sight of the airport and the pilot finds that the
angle of depression to the airport is 25 degrees. Find
the distance between the plane and the airport.
Now we will answer the next question. For the 65° angle, the side
of length 27000 feet is the adjacent side. The side marked as
the distance from a point on the ground directly below the plane
to the airport is the opposite side. A trigonometric function that
has both adjacent and opposite is the tangent function.
opposite
tan( 65) 
adjacent
opposite
tan( 65) 
27000
27000 * tan( 65)  opposite
57901.68685  opposite
angle of elevation = 25°
65°
distance from plane to
airport
27000 feet
airport
distance from a point
on the ground directly
below the plane to the
airport
The distance from a point on the
ground directly below the plane to the
airport is 57901.68685 feet.
A hot-air balloon is floating above a
straight road. To calculate their height
above the ground, the balloonists
simultaneously measure the angle of
depression to two consecutive mileposts
on the road on the same side of the
balloon. The angles of depression are
found to be 23 degrees and 27 degrees.
How high (in feet) is the balloon?
continued on next slide
A hot-air balloon is floating above a straight road. To calculate their height
above the ground, the balloonists simultaneously measure the angle of
depression to two consecutive mileposts on the road on the same side of the
balloon. The angles of depression are found to be 23 degrees and 27
degrees. How high (in feet) is the balloon?
Once again we should draw a picture and label what we know. This
is not drawn to scale.
23
27
A
1 mile
height of the
balloon
B
continued on next slide
A hot-air balloon is floating above a straight road. To calculate their height
above the ground, the balloonists simultaneously measure the angle of
depression to two consecutive mileposts on the road on the same side of the
balloon. The angles of depression are found to be 23 degrees and 27
degrees. How high (in feet) is the balloon?
The height of the balloon is one of the legs of two different right
triangle. One of the triangles is the red one. For this triangle, we
do not know the length of the hypotenuse or the length of the
other leg. We do however know the length of part of the other
leg.
23
27
A
1 mile
height of the
balloon
B
continued on next slide
A hot-air balloon is floating above a straight road. To calculate their height
above the ground, the balloonists simultaneously measure the angle of
depression to two consecutive mileposts on the road on the same side of the
balloon. The angles of depression are found to be 23 degrees and 27
degrees. How high (in feet) is the balloon?
The other of the triangles is the blue one. For this triangle, we do
not know the length of the hypotenuse or the length of the other
leg. You should note though that the length of the other leg is
part of the length of the leg of the red right triangle from the
previous screen.
That part that the
two legs have in
common, we can label
as x.
23
27
A
1 mile
height of the
balloon
B
continued on next slide
A hot-air balloon is floating above a straight road. To calculate their height
above the ground, the balloonists simultaneously measure the angle of
depression to two consecutive mileposts on the road on the same side of the
balloon. The angles of depression are found to be 23 degrees and 27
degrees. How high (in feet) is the balloon?
Now we can work on using a
trigonometric function to
write an equation for the
height of the balloon (we will
label this H). We will need
to know the measure of the
one of the non-right angles.
We will use angle A. Angle A
and the 23° angle add
together to be 90°. This
allows us to find A using A =
90 ° - 23° = 67 °.
A
23
1 mile
A
27
H = height of the
balloon
B x
continued on next slide
A hot-air balloon is floating above a straight road. To calculate their height
above the ground, the balloonists simultaneously measure the angle of
depression to two consecutive mileposts on the road on the same side of the
balloon. The angles of depression are found to be 23 degrees and 27
degrees. How high (in feet) is the balloon?
The two sides of the triangle
that we are interested in are
H (adjacent to A) and the
side opposite angle A. A
trigonometric function that
both opposite and adjacent
is the tangent function.
Thus we can set up the
following:
tan( 67) 
opposite
adjacent
tan( 67) 
1x
H
A
23
A
27
1 mile
H = height of the
balloon
B x
continued on next slide
A hot-air balloon is floating above a straight road. To calculate their height
above the ground, the balloonists simultaneously measure the angle of
depression to two consecutive mileposts on the road on the same side of the
balloon. The angles of depression are found to be 23 degrees and 27
degrees. How high (in feet) is the balloon?
We can now do the same process with the blue right triangle. We
first have to find angle B. Angle B and the 27° angle add up to
90°. Thus we can find B with B = 90 °- 27 °=63 °. Now we can set
up the tangent function expression.
opposite
tan( 63) 
adjacent
23
x
tan( 63) 
H
27
A
1 mile
H = height of the
B balloon
B x
continued on next slide
A hot-air balloon is floating above a straight road. To calculate their height
above the ground, the balloonists simultaneously measure the angle of
depression to two consecutive mileposts on the road on the same side of the
balloon. The angles of depression are found to be 23 degrees and 27
degrees. How high (in feet) is the balloon?
Now we have two equation with two variables. We can solve these
using substitution. Since we are interested in finding H, we should
solve one equation for x and substitute that into the other
equation.
Solve this for x.
x
tan( 63) 
H
H tan( 63)  x
2
3
A
1 mile
27
H = height of the
balloon
x
B
Now substitute this expression for x into the other equation.
tan( 67) 
1  H tan( 63)
H
continued on next slide
A hot-air balloon is floating above a straight road. To calculate their height
above the ground, the balloonists simultaneously measure the angle of
depression to two consecutive mileposts on the road on the same side of the
balloon. The angles of depression are found to be 23 degrees and 27
degrees. How high (in feet) is the balloon?
Finally we solve for H.
1  H tan( 63)
tan( 67) 
H
H tan( 67)  1  H tan( 63)
H tan( 67)  H tan( 63)  1
2
3
H tan( 67)  tan( 63)  1
1
H 
tan( 67)  tan( 63)
H  2.542964269
A
1 mile
27
H = height of the
balloon
x
B
Note that this answer is in the same units as those that your single given
distance is in. That distance is miles. We are asked to find the height in
feet. There are 5280 feet in 1 mile. Thus the answer is
2.542964269*5280=13426.85134 feet.
The angle of elevation to the top
of the Empire State Building in
New York is found to be 11
degrees from the ground at a
distance of 1 mile from the base
of the building. Using this
information, find the height of
the Empire State Building.
continued on next slide
The angle of elevation to the top of the Empire State Building in New
York is found to be 11 degrees from the ground at a distance of 1 mile
from the base of the building. Using this information, find the height of
the Empire State Building.
Once again we should draw a picture and label what we know. This
is not drawn to scale.
Empire State Building
B
11°
Based on this picture we need
to find the length of the side
opposite the angle of elevation
(11°) and we know the length of
the side adjacent to the angle
of elevation. A trigonometric
function that has both the
opposite and adjacent sides is
the tangent function. We can
use this to answer the
question.
1 mile
continued on next slide
The angle of elevation to the top of the Empire State Building in New
York is found to be 11 degrees from the ground at a distance of 1 mile
from the base of the building. Using this information, find the height of
the Empire State Building.
tan( 11) 
B
tan( 11) 
1
tan( 11)  B
Empire State Building
B
opposite
adjacent
B  .1943803091
11°
1 mile
Note that this answer is in the same
units as those that your single given
distance is in. That distance is miles.
We are asked to find the height in feet.
There are 5280 feet in 1 mile. Thus the
answer is
.1943803091*5280=1026.328032 feet