Download 6.7—Trigonometric Ratios

6.7—Trigonometric Ratios
Agile Mind 15
6.7 Warmup
Consider the dimensions shown on the waterslide. Use these dimensions to find
the angle of elevation of the slide, x, and the length of the stairs to the landing
platform, y. (Assume that the 60 ft support makes a right angle with the ground.)
x=
45°
120 ft
y=
y
60 ft
x°
6.7—Trigonometric Ratios
6.7 Day 1 Warm-up
Find the missing side of each triangle. Leave your answers in simplest radical form.
1.
2.
5
7 2
3.
y
y
10
4.
5.
6
4
6
12
6.7—Trigonometric Ratios
6.7 Day 2 Warm-up
Find the missing side of each triangle. Leave your answers in simplest radical form.
1.
2.
3.
8
8
y
4. Find the sinA, cosA
and tanA of ABC
10
5.
A
10
C
3
4
1
B
5
10
6.7—Trigonometric Ratios
Agile Mind 15
cosine and the _________
tangent of an acute angle.
Objective: Find the _____,
sine the _______,
1a. What can you say about ΔBCD, ΔGCE, and ΔACF?
1b. Are the triangles in any set of nested right triangles
always similar? Why or why not?
2a. What is the name of the ratio of the length of the opposite leg to the
length of the adjacent leg of an acute angle in a right triangle?
2b. What is the relationship between the tangent of an acute angle and
the slope of the hypotenuse?
5. You have defined the tangent of an acute angle. What are the names
and definitions of the other two important ratios in a right triangle?
two sides
A trigonometric ratio is a _______
________ of a right
ratio of the lengths of ______
cos and
triangle. The three basic trigonometric ratios are sine (____),
sin cosine (_____),
tangent (_____).
Let ΔABC be a right triangle. The sine, the cosine, and the
tan
tangent of the acute angle A are defined as follows.
sin 𝐴 =
side opposite
hypotenuse
side adjacent
cos 𝐴 =
hypotenuse
side opposite
tan 𝐴 =
side adjacent
∠𝐴
a
=
c
𝑡𝑜 ∠𝐴
b
=
c
∠𝐴
a
=
𝑡𝑜 ∠𝐴
b
Soh Cah Toa
6. Complete the following trigonometric ratios.
REINFORCE
In the right triangle shown, find the tangent of ∠A and the tangent of ∠B.
How are these two ratios related to each other?
In the right triangle shown, find the sine and cosine of ∠A and the sine and
cosine of ∠B. How are these two ratios related to each other?
Examples:
Find the sine, cosine, and tangent of the acute angles of the triangle. Express each value as a simplified exact
answer and then as a decimal rounded to four places.
1.
2.
Use a calculator to approximate the given value to four decimal places.
3.
tan 23°
4.
sin 56°
5.
cos 63°
10. John is flying a kite and has let out 100 feet of string. He is holding the
string 5 feet above the ground at an angle of 40° with the horizontal. Fill in the
blanks to find the approximate altitude of the kite.
The kite is ____________
above the ground.
Find the value of each variable. Give answers in exact form and then use a calculator to give the answers as
decimal approximations rounded to the nearest tenth.
6.
7.
7. Based on your knowledge of special right triangles, fill in the chart
below.
8a. Find
on your calculator to the nearest thousandth.
8b. Use your calculator to find sin 60° to the nearest thousandth.
9. Earlier you solved the tree stabilization
problem using special right triangles. Now
use trigonometric ratios to find how high
on the tree to attach the wires and how
long the wires need to be.
 The angle that your line of sight makes with a line drawn
angle ___
of ___________.
elevation
horizontally is called the __________
8. If the angle of elevation from your position on the ground to the top
of a building is 67° and you are standing 30 meters from the foot of
the building, approximate the height of the building.
9. Find the perimeter of rectangle MNOP to the nearest tenth of a
centimeter.