Transcript Trigonometry

The Tangent Ratio
Lesson 8-3
Geometry
Additional Examples
Write the tangent ratios for
A and
B.
opposite
BC
20
=
=
=
tan A adjacent
AC
21
opposite
AC
21
=
=
=
tan B adjacent
BC
20
The Tangent Ratio
Lesson 8-3
Geometry
Additional Examples
To measure the height of a tree, Alma walked 125 ft from the
tree and measured a 32° angle from the ground to the top of the tree.
Estimate the height of the tree.
The tree forms a right angle with the ground, so
you can use the tangent ratio to estimate the
height of the tree.
height
tan 32° = 125
height = 125 (tan 32°)
125
32
78.108669
Use the tangent ratio.
Solve for height.
Use a calculator.
The tree is about 78 ft tall.
The Tangent Ratio
Lesson 8-3
Geometry
Additional Examples
Find m R to the nearest degree.
47
tan R = 41
m R
tan–1
47
41
So m R
Find the tangent ratio.
47
41
48.900494
49.
Use the inverse of the tangent.
Use a calculator.
Sine and Cosine Ratios
Lesson 8-4
Geometry
Additional Examples
Use the triangle to find sin T, cos T, sin G, and cos G. Write
your answer in simplest terms.
12
3
opposite
=
=
sin T = hypotenuse
20
5
16
4
adjacent
=
=
cos T = hypotenuse
20
5
16
4
opposite
=
=
sin G = hypotenuse
20
5
12
3
adjacent
=
=
cos G = hypotenuse 20
5
Sine and Cosine Ratios
Lesson 8-4
Geometry
Additional Examples
A 20-ft. wire supporting a flagpole forms a 35˚ angle with the
flagpole. To the nearest foot, how high is the flagpole?
The flagpole, wire, and ground form a right triangle with
the wire as the hypotenuse.
Because you know an angle and the measures of its
adjacent side and the hypotenuse, you can use the
cosine ratio to find the height of the flagpole.
height
cos 35° = 20
height = 20 • cos 35°
20
35
16.383041
Use the cosine ratio.
Solve for height.
Use a calculator.
The flagpole is about 16 ft tall.
Sine and Cosine Ratios
Lesson 8-4
Geometry
Additional Examples
A right triangle has a leg 1.5 units long and hypotenuse 4.0
units long. Find the measures of its acute angles to the nearest degree.
Draw a diagram using the information given.
Use the inverse of the cosine
function to find m A.
1.5
cos A = 4.0 = 0.375
Use the cosine ratio.
m A = cos–1(0.375)
Use the inverse of the cosine.
0.375
67.975687
m A
68
Use a calculator.
Round to the nearest degree.
Sine and Cosine Ratios
Lesson 8-4
Geometry
Additional Examples
(continued)
To find m B, use the fact that the acute angles of a right triangle are
complementary.
m A + m B = 90
68 + m B
90
m B
22
Definition of complementary angles
Substitute.
The acute angles, rounded to the nearest degree, measure 68 and 22.
Angles of Elevation and Depression
Lesson 8-5
Geometry
Additional Examples
Describe
1 and
2 as they relate to the situation shown.
One side of the angle of depression is a horizontal line.
1 is the angle of depression from the airplane to the building.
One side of the angle of elevation is a horizontal line.
2 is the angle of elevation from the building to the airplane.
Angles of Elevation and Depression
Lesson 8-5
Geometry
Additional Examples
A surveyor stands 200 ft from a building to measure its height
with a 5-ft tall theodolite. The angle of elevation to the top of the
building is 35°. How tall is the building?
Draw a diagram to represent the situation.
x
tan 35° = 200
Use the tangent ratio.
x = 200 • tan 35°
200
35
140.041508
So x
Solve for x.
Use a calculator.
140.
To find the height of the building, add the height of the Theodolite,
which is 5 ft tall.
The building is about 140 ft + 5 ft, or 145 ft tall.
Angles of Elevation and Depression
Lesson 8-5
Geometry
Additional Examples
An airplane flying 3500 ft above ground begins a 2° descent
to land at an airport. How many miles from the airport is the airplane
when it starts its descent?
Draw a diagram to represent the situation.
sin 2° =
x=
3500
2
5280
100287.9792
18.993935
3500
x
3500
sin 2°
Use the sine ratio.
Solve for x.
Use a calculator.
Divide by 5280 to convert
feet to miles.
The airplane is about 19 mi from the airport when it starts its descent.