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Pretest
Please complete the pretest for this standard
on your own. Try to remember all you can from
our first discussion of this topic.
Define and use the trigonometric
ratios (sine, cosine, tangent,
cotangent, secant, cosecant) in
terms of angles of right triangles.
MA.912.T.2.1
LEQ: How do I use trigonometric ratios to
find angles and sides of right triangles?
Vocabulary
Matching
C
A. The side opposite the right
angle of a right triangle
D
B. The ratio of the side opposite
the angle to the side adjacent to
the angle
_____ sine of an angle
_____ cosine of an angle
B
_____ tangent of an angle
C. The ratio of the side opposite
the angle to the hypotenuse of
the triangle
A
_____ hypotenuse
D. The ratio of the side adjacent
the angle to the hypotenuse of
the triangle
I do…
There are two types of trig problems.
2.
You have to find the angle given two
sides of a right triangle. In this case
you need to use the inverse trig
function. This is found by using the
“second” or “shift” key on your
calculator.
Basic Trigonometric Ratios
1.
You have to find a side given another
side and an angle. In this case, you
will calculate the value of the trig
function evaluated at the given angle
and then solve for the missing side.
I do…
Refer to right triangle ABC.
5
,
12
what is cosine of ÐA?
13
5
opposite 5
tan B =
=
adjacent 12
In order to find the cosine, we need
to know the hypotenuse. You may
recognize that this is a Pythagorean
triple, and that the hypotenuse is
13. Otherwise, you can apply the
Pythagorean Theorem to find the
hypotenuse.
12
cos A =
opposite
hypotenuse
cos A =
12
13
Basic Trigonometric Ratios
If the tangent of ÐB is
I do…
If the object is below the level of the observer, then the angle between the
horizontal and the observer's line of sight is called the angle of depression.
horizontal
Angle of depression
Notice that the angle of
elevation and the angle of
depression are alternate
interior angles and are
EQUAL!
Angle of elevation
horizontal
Angles of Elevation and Depression
The angle of elevation of an object as seen by an observer is the angle
between the horizontal and the line from the object to the observer's eye
(the line of sight).
I do…
Draw a picture.
Set up a ratio and solve.
sin x =
opposite
hypotenuse
40 feet
sin33 =
x
40
x
40sin33 = x
x » 21.8
33 degrees
Draw a picture to solve a word problem
A kite is held by a taut string pegged to the ground. The string is
40 feet long and makes a 33 degree angle with the ground.
Supposing that the ground is level, find the vertical distance from
the ground to the kite.
I do…
The driver of the boat wants to
change direction to sail toward the
restaurant. Which of the following
is closest to the value of x?
Real World Application
A tackle shop and restaurant are located on the shore of a lake and are 32
meters (m) apart. A boat on the lake heading toward the tackle shop is a
distance of 77 meters from the tackle shop. This situation is shown in the
diagram below, where point T represents the location of the tackle shop, point R
represents the location of the restaurant, and point B represents the location
of the boat.
This problem has a
LOT of text, but it
really is just a basic
trig ratio problem.
The hypotenuse is not labeled in this
problem, only the sides opposite and
adjacent to x. This must be a tangent
problem.
tan x =
opposite
adjacent
32
tan x =
77
æ 32 ö
x = tan ç ÷
è 77 ø
-1
x » 22.6
I do…
In this problem, we are given an angle,
and asked to find a side, so we will not
need to use an inverse function.
The trig function that uses
opposite and hypotenuse
is sine.
opposite
sin x =
hypotenuse
sin 25 =
6
.
6
Hypotenuse = x
x » 16.6
7
7
x=
x
sin 25
1
opposite
Real World Application
Mr. Rose is remodeling his house by adding a room to one side, as shown
in the diagram below. In order to determine the length of the boards he
needs for the roof of the room, he must calculate the distance from point
A to point D.
We do
• Sage and Scribe
We do
Pick a Card
With your partner, takes turns picking a card.
The student that picked the card shares it with
their partner and offers the explanation or
solution. The other student verifies for
accuracy, then picks a card to take their turn.
You Do
1.
Find the value of w, then x.
Round length to the nearest
tenth and angle measures
to the nearest degree.
2.
Find the angle of elevation of the sun from the ground to the top of a tree
that is 10 yards tall casts a shadow 14 years long. Round to the nearest
degree.
3.
Find the value of x.
You Do
1.
Find the value of w, then x.
Round length to the nearest
tenth and angle measures
to the nearest degree.
sin 50 =
w
10
10sin50 = w
w » 7.7
7.7
sin x =
11
-1 æ 7.7 ö
x = sin ç ÷
è 11 ø
æ 7.7 ö
x = sin ç ÷
è 11 ø
-1
x » 44°
You Do
2.
Find the angle of elevation of the sun from the ground to the top of a tree
that is 10 yards tall casts a shadow 14 years long. Round to the nearest
degree.
10
tan x =
14
æ 10 ö
x = tan ç ÷
è 14 ø
10
-1
x » 36°
14
You Do
3.
Find the value of x.
The angle of elevation is
also 10. We use the sine
function.
sin10 =
x=
200
x
200
sin10
x » 1151.8
Post Test
Please complete the post test for this standard
on your own. Do the best you can, hopefully
you will show improvement over your pretest
score.