Download Packet 5 for Unit 5 M2G

M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
ASSIGNMENT SHEET FOR PACKET 5 OF UNIT 5
This packet includes sections 8-4 from our textbook and the review for that section.
Date Due
Number
Assignment
5N
p. 573 # 17, 21
5O
p. 574 # 28 – 34 all
5P
p. 574-575
# 36 – 41 all, 46
Topics
8-4 Trigonometry
Vocabulary: trigonometric ratio, sine,
cosine, tangent
Find the sine, cosine, or tangent of an
angle given side lengths in a right
triangle
Use sine, cosine, and tangent or their
inverses to solve problems involving right
triangles
8-4 Trigonometry
Vocabulary: angle of elevation, angle
of depression
Use sine, cosine, and tangent to solve
problems involving right triangles
8-4 Trigonometry
Vocabulary: inverse sine, inverse cosine,
inverse tangent
Find the sine, cosine, or tangent of an
angle given side lengths in right triangle
Use inverse sine, cosine, and tangent to
solve problems involving right triangles
Quiz on 8-4
1
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
INVESTIGATION: INTRODUCTION TO TRIGONOMETRIC RATIOS
1. Find the missing side lengths in the 45-45-90 triangle shown
at right, and label them in the diagram.
B
2. Write “hyp” next to the hypotenuse in the diagram.
3. The side adjacent to ∠A is the leg that has A as its
endpoint. Write “adj” next to side adjacent to ∠A in the
diagram.
4. The side opposite from ∠A is the leg that does not touch
point A. Write “opp” next to the side opposite from ∠A in
the diagram.
10
45°
C
A
5. Find the following ratios. Round answers to 3 decimal places, if necessary.
opposite
=
hypotenuse
adjacent
=
hypotenuse
opposite
=
adjacent
6. Use a calculator to find the following, rounded to 3 decimal places if necessary. Before
you start, press MODE, and make sure that DEGREE instead of RADIAN has been selected.
sin ( 45 )
cos ( 45 )
tan ( 45 )
B
7. Repeat questions 1 through 5 for the 30-60-90 triangle shown at right.
15
A
opposite
=
hypotenuse
adjacent
=
hypotenuse
opposite
=
adjacent
8. Use a calculator to find the following, accurate to 3 decimal places.
sin ( 30 )
cos ( 30 )
tan ( 30 )
9. What do you notice?
2
30°
C
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
The ratio of the lengths of two sides of a right triangle is a trigonometric ratio. The three most
common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan.
sin A =
leg opposite ∠A
hypotenuse
cos A =
leg adjacent to ∠A
hypotenuse
tan A =
leg opposite ∠A
leg adjacent to ∠A
You can use the acronym SOH CAH TOA to remember these formulas. They mean:
Sin ( angle ) = Opposite
Hypotenuse
Cos ( angle ) = Adjacent
Hypotenuse
Tan ( angle ) = Opposite
Adjacent
Ex. 1: Find sin A, cos A, and tan A. Express each ratio as a
fraction in simplest form.
Solution: Label the sides “hyp”, “opp”, and “adj”.
Then fill in the ratios using SOH CAH TOA:
sin (=
A)
opp 5
=
hyp 13
cos (=
A)
adj 12
=
hyp 13
opp
tan (=
A)
opp 5
=
adj 12
hyp
adj
10. How would the answers in Example 1 change if we found sin B, cos B, and tan B instead?
Try this:
11. Find sin J, cos J, tan J, sin L, cos L, and tan L. Express each ratio as a
fraction in simplest form.
3
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
APPLICATIONS OF SINE, COSINE, AND TANGENT
SOH CAH TOA can be used to write trigonometric equations that help us solve for side
lengths or angle measures in a right triangle.
When x is in the numerator:
1. Follow the steps to find the length of side x:
a. Label the sides of the right triangle hyp, opp, and adj, as they
relate to the 50 ° angle.
b. Which part of SOH CAH TOA uses the sides with length x and
12? Use this part of SOH CAH TOA to write an equation about
50 ° , x, and 12.
______ (
x
50 °
12
) = ______
c. Multiply to solve for x. Before using your calculator, press MODE, and make sure it is set
to DEGREE, not RADIAN, mode. Round to the nearest tenth.
When x is in the denominator:
x
2. Follow the steps to find the length of side x:
a. Label the sides of the right triangle hyp, opp, and adj, as they
relate to the 25 ° angle.
25
32
b. Which part of SOH CAH TOA uses the sides with length x and
32? Use this part of SOH CAH TOA to write an equation about 25 ° , x, and 32.
______ (
) = ______
c. To solve, draw a fraction bar under the left side, and give it a denominator of 1. Find
the cross-products; set them equal to each other, and divide to solve for x. Round to
the nearest tenth.
4
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
3. Try these. Round answers to the nearest tenth.
a.
b.
x
35 °
10
9
x
38 °
Angle of Elevation and Angle of Depression
Angles of elevation and depression are non-right angles formed along the horizontal side of
a right triangle.
4. Follow the steps to solve this problem: The angle of elevation from a point on the ground
to the top of a tree is 20° . If the point on the ground is 13 feet from the base of the tree,
find the height of the tree to the nearest tenth of a foot.
a. Draw a right triangle.
b. Write “ 20° ” in the angle formed by the horizontal side of the triangle. This is the angle
of elevation.
c. Write “x” along the side of the triangle that represents the height of the tree. This is
what we’re being asked to find.
d. Write “13 ft” along the side of the triangle that represents the distance from the point
on the ground to the base of the tree.
e. Use SOH CAH TOA to write a trigonometric equation about 20° , x, and 13. Then solve,
and answer the problem using the appropriate units and rounding.
5
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
Another Application of Trigonometry
5. A ladder rests against a building at a point 6.3 ft from the ground. If the ladder makes a
47° angle with the ground, how long is the ladder? Round your answer to the nearest
tenth.
a. Draw a right triangle representing the building, the ladder, and the ground.
b. Write 6.3 ft, 47° , and x in the appropriate sections of your diagram.
c. Use SOH CAH TOA to write a trigonometric equation about 6.3 ft, 47° , and x, and
solve. Then answer the problem using the appropriate units and rounding.
6. Isolate x in each equation. Do not solve.
x
a. tan ( 35° ) =
12
22
b. sin ( 83° ) =
x
6
102
c. cos (18° ) =
x
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
Inverse Trigonometric Ratios
With a calculator and sine, cosine, or tangent, you can find the measure of an angle in a
right triangle, using the inverse of the trigonometric ratio.
1. Follow the steps to find the measure of angle x:
a. Label the sides of the right triangle hyp, opp, and adj, as
they relate to angle x.
7.25
4.37
b. Which part of SOH CAH TOA uses the 4.37 and 7.25? Use
this part of SOH CAH TOA to write an equation about x, 4.37, and 7.25.
______ (
x
) = ______
c. To solve for an angle measure, use an inverse trigonometric function. In a sine
equation, take the inverse sine of both sides (on your calculator, 2ND, SIN); in a cosine
equation, use inverse cosine, and in a tangent equation, use inverse tangent.
Try this with your equation in part (b). Round your answer to the nearest tenth, and
label with the degree symbol.
Notice that inverse sine appears as sin −1 on your calculator.
2. Rewrite each sine, cosine, or tangent equation as a corresponding equation using inverse
sine, inverse cosine, or inverse tangent. Write sin −1 for inverse sine, cos −1 for inverse cosine,
or tan −1 for inverse tangent.
a. sin ( x ) =
8.2
12.5
b. cos ( x ) =
82
97
7
c. tan ( x ) =
13
5
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
3. Try these. Round answers to the nearest tenth. Label angle measures with the degree
symbol.
a.
b.
x
11
12
x
6.5
4.2
4. Follow the steps to solve this problem: A camera is mounted on a wall 7.4 feet directly
above a security desk. The security desk is 9.3 feet away from a door. To the nearest tenth
of a degree, find the angle of depression from the camera to the door.
a. Draw a right triangle so that its horizontal leg is at the top of your diagram.
b. Write “x” in the angle formed by the horizontal side of the triangle. This is the angle of
depression, and it’s what we’re being asked to find.
c. Write “7.4 ft” along the side of the triangle that represents the height of the camera
above the security desk.
d. Write “9.3 ft” along the side of the triangle that represents the distance from the
security desk to the door.
e. Use SOH CAH TOA to write a trigonometric equation about x, 7.4, and 9.3. Then solve
your equation, and answer the problem using the appropriate units and rounding.
8
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
REVIEW FOR SECTION 8-4
Things to remember:
sin ( angle ) =
opposite
hypotenuse
cos ( angle ) =
adjacent
hypotenuse
tan ( angle ) =
opposite
adjacent
•
When x is in the denominator, write a 1 under the left side. Then set the cross-products equal to each
other, and divide to solve.
•
When x is in the numerator, multiply both sides by the denominator to solve.
•
•
 opposite 
−1  opposite 
−1  adjacent 
When x is the angle, use sin −1 
 . Type
 , cos 
 , or tan 
 hypotenuse 
 hypotenuse 
 adjacent 
inverse trig functions using 2ND, SIN, COS, or TAN on your calculator.
Angles of elevation or depression always border the horizontal leg of a right triangle.
1. Find the length of the missing side in each triangle below, and then find the requested trigonometric ratios.
Figures may not be drawn to scale. Give answers as reduced fractions.
B
a. sin A =
E
f. cos D =
17
b. cos A =
c. tan B =
15
g. tan N =
12
h. cos K =
i. tan D =
e. sin N =
j. sin E =
D
F
K
A
d. sin D =
8
C
12
M
16
2. Solve for x in each equation. Show your work. Round answers to one decimal place.
82
x
100
8
a. sin 68° =
b. tan 22° =
c. cos θ =
d. sin10° =
x
x
11
12
9
N
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
3. Write a trigonometric equation, and solve to find the value of x. Round answers to the nearest hundredth if
necessary.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Xo
Xo
Xo
10
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
4. For each, (a) draw and label a right triangle using x and the given quantities. (b) Then write and solve a
trigonometric equation, and (c) follow the rounding directions, and give your answer with appropriate units.
a. Matt measured the angle of elevation from a position 43 meters away from base of a rock formation to
the top of the rock formation. He found the angle to be 36° . Find the height of the rock formation to the
nearest meter.
Draw and label a diagram:
Write equation, and solve:
Answer:
b. Jess is standing in view of a 200-foot-high radio tower. From where she stands, the angle of elevation to
the top of the tower is 49° . To the nearest foot, find the distance between Jess and the base of the tower.
Draw and label a diagram:
Write equation, and solve:
Answer:
c. A 60-foot ramp rises from the first floor to the second floor of a parking garage. The ramp makes a 15°
angle with the ground. How high above the first floor is the second floor? Round your answer to the
nearest tenth.
Draw and label a diagram:
Write equation, and solve:
Answer:
11
M2 GEOMETRY PACKET 5 FOR UNIT 5 – SECTION 8-4 TRIGONOMETRY
d. A kite is tied to the ground with a 150-foot string. If the angle of depression from the kite to the point at
which it is tied to the ground is 40° , how high above the ground is the kite? Round your answer to the
nearest tenth.
Draw and label a diagram:
Write equation, and solve:
Answer:
e. A plane is flying at an altitude of 12,000 meters. The distance from a point along the ground directly
below the plane and the base of a control tower is 19,200 meters. Find the angle of depression between
the plane and the base of the control tower. Round to the nearest tenth.
Draw and label a diagram:
Write equation, and solve:
Answer:
f. A shark swims 22 feet below sea level. If the angle of depression from a boat at sea level and the shark
is 20° , find the most direct distance between the boat and the shark, rounded to the nearest tenth.
Draw and label a diagram:
Write equation, and solve:
Answer:
12