Download Trigonometry packet Geometry honors

Hertel/Williams/Lambert
Trigonometry packet
Geometry honors
Table of contents:
Day 1: Basic Trigonometry Review
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use
trigonometric ratios and inverse trigonometric relations to find missing angles.
Day 2: Trig Review and Co-Functions
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine
in terms of its CoFunction.
Day 3: Using Trigonometry to Determine Area
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
Day 4: Law of Sines
SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures
of two angles.
Day 5: Law of Cosines
SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure
of the included angle.
Day 6: Review
Day 7: TEST
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and
inverse trigonometric relations to find missing angles.
Day 1
Basic Trigonometry Review
Warm Up: Review the basic Trig Rules below and complete the example below:
Basic Trigonometry Rules:

These formulas ONLY work in a right triangle.

The hypotenuse is across from the right angle.

Questions usually ask for an answer to the nearest units.

You need a scientific or graphing calculator.
Example:
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and
inverse trigonometric relations to find missing angles.
Example 1:
Practice:
1.)
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and
inverse trigonometric relations to find missing angles.
2.)
Example 2:
Example 3: Inverse Trigonometric Relations to find Missing Angles.
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and
inverse trigonometric relations to find missing angles.
Practice:
3.)
4.)
5.)
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and
inverse trigonometric relations to find missing angles.
Exit Ticket:
Summary:
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and
inverse trigonometric relations to find missing angles.
Day 1 Homework:
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and
inverse trigonometric relations to find missing angles.
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and
inverse trigonometric relations to find missing angles.
8.
10.
9.
11.
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
Day 2
Basic Trig Word Problems and CoFunctions
Warm Up:
1.) Find the length of side p and the measure of angle m, as shown on the diagram. Give each
answer correct to the nearest whole number or degree.
2.)
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
Real World Connection:
Trigonometry can be used on a daily basis in the workplace. Since trigonometry
means "triangle measure", any profession that deals with measurement deals with
trigonometry as well. Carpenters, construction workers and engineers, for example,
must possess a thorough understanding of trigonometry.
Angle of Elevation
The angle of elevation is always measured from the
ground up. Think of it like an elevator that only goes
up. It is always INSIDE the triangle.
In the diagram at the left, x marks the angle of elevation
of the top of the tree as seen from a point on the ground.
You can think of the angle of elevation in relation to the
movement of your eyes. You are looking straight ahead
and you must raise (elevate) your eyes to see the top of
the tree.
Angle of Depression
The angle of depression is always OUTSIDE the
triangle. It is never inside the triangle.
In the diagram at the left, x marks the angle of depression
of a boat at sea from the top of a lighthouse.
You can think of the angle of depression in relation to the
movement of your eyes. You are standing at the top of
the lighthouse and you are looking straight ahead. You
must lower (depress) your eyes to see the boat in the
water.
There are two possible ways to use our angle of depression to
obtain an angle INSIDE the triangle.
1.
2.
Find the angle adjacent (next door) to our angle which
is inside the triangle. This adjacent angle will always
be the complement of our angle. Our angle and the
angle next door will add to 90º. In the diagram on the
left, the adjacent angle is 55º.
Utilize the fact that the angle of depression = the
angle of elevation and simply place 35º in angle
A. (the easiest method) Just be sure to place it in the
proper position.
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
Example 1:
Practice:
1.) From a point on the ground 25 feet from the foot of a tree, the angle of elevation of the top
of the tree is 32º. Find to the nearest foot, the height of the tree.
2.)
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
3.)
Example 2:
Refer to the triangle below:
a) What is the relationship between mA and mB? ____________________
A
13
12
C
5
b) What is the cos A? _____ What is the sine B? _____
B
c) What is the sin A? _____ What is the cos B? _____
What do you notice about the cosine and sine of complements?
_____________________________________________________________
The sine of an acute angle is equal to the cosine of its complement.
The sine and cosine functions are called cofunctions.
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
Any trigonometric function of an acute angle is equal to the cofunction of its complement.
cos x = sin (90o - x)
sin x = cos (90o - x)
sec
= csc (90o - )
cscof
 =sin
sec 30
(90oo.- )
Forexample,
find the value
Now find the value of cos 60o.
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
Exit Ticket:
Summary:
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
Homework:
1.
Find the value of
a.
(
)
b.
(
(
)
c.
d.
2.
3.
that makes each statement true.
(
)
)
a.
Make a prediction about how the sum
b.
Use the sine and cosine values of special angles to find the sum:
c.
Find the sum
d.
Was your prediction a valid prediction? Explain why or why not.
Langdon thinks that the sum
sum of the sines of angles.
4.)
5.)
6.)
will relate to the sum
.
.
.
is equal to
. Do you agree with Langdon? Explain what this means about the
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
Summary:
SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms
of its CoFunction.
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
Day 3
Using Trigonometry to Determine Area
Warm Up:
Determine the Area for each triangle below, if possible. If not possible, state what
additional information would be needed.
Triangle #1
Triangle #2
Triangle #3
*If there is missing information, is there a way to find the missing information?
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
Example 1:
What if the third side length of the triangle were provided? Is it possible to determine
the area of the triangle now? Find the area of
. (Hint: Draw Altitude HJ)
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
Discussion:
Let’s look at
which is set up similarly to the triangle in Example 1:
What formula would represent the area of this triangle?
A= ½ ah
Let’s rewrite the equation. Describe what you see:
A= ½ ah
_________________________________
A= ½ ab
_________________________________
Look at
below with an arc mark at vertex
labeled . What do you notice
about the structure of ? Can we think of this newly written area formula in a
different way using trigonometry?
Area of a Triangle:
(Circle the part that represents the height)
Using the new formula, what information would you need to calculate the area of a
triangle?
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
Example 2:
A farmer is planning how to divide his land for planting next year’s crops. A triangular plot of
land is left with two known side lengths measuring
and
. How could the
farmer calculate the area of the plot, if the included angle between the known side lengths is
30⁰?
You Try It:
A real estate developer and her surveyor are searching for their next piece of land to build on. They
each examine a plot of land in the shape of
. The real estate developer measures the length of
and
and finds them to both be approximately
feet, and the included angle has a
measure of approximately
. The surveyor measures the length of
and
and finds the lengths
to be approximately
feet and
feet, respectively, and measures the angle between the two
sides to be approximately
.
a. Draw a diagram that models the situation, labeling all lengths and angle measures.
b. The real estate developer and surveyor each calculate the area of the plot of land and
both find roughly the same area. Show how each person calculated the area; round to
the nearest hundred.
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
Challenge:
In isosceles triangle
, the base
, and the base angles have measures of
. Find the area of
.
Summary:
We have just discovered that the area of a triangle can be expressed using the lengths of two sides
and the sine of the included angle. This is often referred to as the SAS Formula for the area of a
triangle.
The "letters" in the formula may change from problem to problem, so try to remember the pattern
of "two sides and the sine of the included angle".
We no longer have to rely on a problem supplying us with the length of the altitude (height) of
the triangle in order for us to find the area of the triangle. If we know two sides and the
included angle, we are in business.
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
Exit Ticket:
Given two sides of the triangle shown, having lengths of
and , and their included angle of
Homework:
Find the area of each triangle. Round each answer to the nearest tenth.
1.
2.
3.
, find the area of the triangle.
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
4.
5.
In
,
,
, and
. Determine the area of the triangle. Round to the nearest tenth.
6.
A landscape designer is designing a flower garden for a triangular area that is bounded on two sides by the client’s house and
driveway. The length of the edges of the garden along the house and driveway are
and
respectively, and the edges
come together at an angle of
. Draw a diagram, and then find the area of the garden to the nearest square foot.
7.
A right rectangular pyramid has a square base with sides of length . Each lateral face of the pyramid is an isosceles triangle.
The angle on each lateral face between the base of the triangle and the adjacent edge is
. Find the surface area of the
pyramid to the nearest tenth.
8.
The Pentagon Building in Washington D.C. is built in the shape of a regular pentagon. Each side of the pentagon measures
in length. The building has a pentagonal courtyard with the same center. Each wall of the center courtyard has a length
of
. What is the approximate area of the roof of the Pentagon Building?
SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.
9.
A regular hexagon is inscribed in a circle with a radius of . Find the perimeter and area of the hexagon.
10. In the figure below,
11. Let
is acute. Show that
.
be the measure of the acute angle formed by diagonals ̅̅̅̅ and ̅̅̅̅ . Show that
be a quadrilateral. Let
.
(Hint: Apply the result from Problem 10 to
and
.)
SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.
Day 4
Law of Sines
Warm Up:
1.) Find the lengths of
and .
2.) Find the lengths of
and . How is this different from part (a)?
Law of Sines:
Today we will show how to find unknown measurements in triangles that are not right triangles;
we can use these formulas for acute and obtuse triangles, but today we will specifically study acute
triangles. The two formulas are the Law of Sines and the Law of Cosines. Today we will work with
the Law of Sines.
 LAW OF SINES: For ALL TRINGLES. Given
with angles
sides opposite them , , and , the Law of Sines states:
,
, and
and the
SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.
The ratio of the sine of an angle in a triangle to the side
opposite the angle is the same for each angle in the triangle.
*The objective of being able to state the law in words is to move the focus away from specific letters and
generalize the formula for any situation.
Proof:
Consider
with an altitude drawn from
a) What is Sin
o Sin
to
.
?
=
o h=
b) What is Sin
o Sin
?
=
o h=
How can we use the information above to prove a part of the Law of
Sines?
SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.
Example 1:
A surveyor needs to determine the distance between two points and that lie on opposite banks of
a river. A point is chosen
meters from point , on the same side of the river as . The
measures of angles
and
are
and
, respectively. Approximate the distance from
to to the nearest meter. Use the Law of Sines to set up all possible ratios.
Practice:
1.) In
,
,
, and
. Find
. Include a diagram in your answer.
2.) A car is moving towards a tunnel carved out of the base of a hill. As the accompanying
diagram shows, the top of the hill, , is sighted from two locations, and . The distance
between and is
. What is the height, , of the hill to the nearest foot?
SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.
More Practice:
3.)
4.) Solve for x:
Challenge:
SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.
Summary:
Exit Ticket:
In the accompanying diagram of
What is the length of side
,
,
, and
.
to the nearest tenth?
1) 6.6
3) 11.5
2) 10.1
4) 12.0
Homework:
1.
Given
,
,
, and
, calculate the
measure of angle to the nearest tenth of a degree, and use the
Sines to find the lengths of
and
to the nearest tenth.
Calculate the area of
2.
Law of
to the nearest square unit.
Given
,
, and
, calculate the measure of
, and use
of Sines to find the lengths of ̅̅̅̅ and ̅̅̅̅ to the nearest hundredth.
the Law
SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.
3.
Does the law of sines apply to a right triangle? Based on
, the following ratios were set up according to the law of sines.
Fill in the partially completed work below.
What conclusions can we draw?
4.
Given quadrilateral
,
,
,
,
is a right angle and
find the length of
, and then find the lengths of ̅̅̅ and ̅̅̅ to the nearest tenth of an inch.
, use the law of sines to
5.
To find the distance across a canyon, a surveying team locates points A and B on one side of the canyon and
point C on the other side of the canyon. The distance between points A and B is 85 yards. The measure of 
SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.
CAB = 68°, and the measure of  ABC = 75°. What is the distance across the canyon?
6.
A surveyor walks 380 meters from point A to point B. Then the surveyor turns 80° to the left and walks 240 meters to point C,
as in the figure below. What is the approximate length of the marsh?
SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.
Day 5
Law of Cosines
Warm Up:
A man is standing 100 feet away from the building, he notices that the angle of
elevation to the top of the building is 41º and that the angle of elevation to a poster on
the side of the building is 21º. How far is the poster from the roof of the building?
Law of Cosines:
In the last lesson we explored one formula that showed us how to find unknown measurements in
triangles that are not right triangles. Today we will learn another one, called the Law of Cosines.

LAW OF COSINES: For ALL TRINGLES.
with angles
opposite them , , and , the law of cosines states:
,
, and
and the sides
.
The square of one side of the triangle is equal to the sum of
the squares of the other two sides minus the twice the
product of the other two sides and the cosine of the angle
between them.
*The objective of being able to state the law in words is to move the focus away from specific letters and
generalize the formula for any situation.
SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.
The Law of Cosines is really just the Pythagorean Theorem, with a something extra so
it works for all triangles. Let’s test this out.
Try it when the
 What happens to the right side of the equation?
________________________________________________________________
________________________________________________________________
Let’s look back at
from the last lesson, with an altitude drawn from to
,
but this time the point where the altitude meets
, point , divides
into lengths
and .
Express e and h using Trigonometry with respect to
o h=
o e=
.
SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.
Now we turn to right triangle
the triangle?

. What length relationship can be concluded between the sides of
By the Pythagorean theorem, the length relationship in
Substitute the trigonometric expressions for
and
into this statement. (b = e + d)
After simplifying this equation as much as possible we end up with:
Other arrangements of this formula:


is
SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.
Example 1:
Our friend the surveyor from yesterday is doing some further work. He has already
found the distance between points and (132 m). Now he wants to locate a point
that is equidistant from both and and on the same side of the river as . He has
his assistant mark the point so that the angles
and
both measure
.
What is the distance between and to the nearest meter?
Practice 1:
Parallelogram
has a measure of
has sides of lengths
and
, and one of the angles
. Approximate the length of diagonal
to the nearest
.
SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.
Challenge:
Summary:
Exit Ticket:
SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.
Homework:
1.
Given
, use the law of cosines to find the length of the side marked
2.
Given triangle
,
3.
Given triangle
length of ̅̅̅̅ .
,
,
, and
,
Calculate the area of triangle
, and
.
to the nearest tenth.
, use the law of cosines to find the length of ̅̅̅̅̅ to the nearest tenth.
. Draw a diagram of triangle
, and use the law of cosines to find the
SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.