Transcript The Tangent Ratio

CHAPTER 7
RIGHT TRIANGLE TRIGONOMETRY
The Tangent Ratio
You will need a protractor for this activity
1. Each person draw a right triangle (∆ABC)
where ‫ﮮ‬A has a measure of 30º.
2. Each person in the group should draw
the triangle with different side lengths, then
measure the legs using inches.
3. Compute the ratio leg opposite ‫ﮮ‬A
leg adjacent ‫ﮮ‬A
4. Compare the ratio with the others in the
group. Make a conjecture.
Trigonometry and the Tangent Ratio
Objectives:
• Use tangent ratios to determine side lengths in
triangles
Previously, to find measures in a right triangle, we used:
• Pythagorean Theorem
• Distance Formula
• 30-60-90 or 45-45-90 special right triangles theorems
Now, we will use Trigonometry (triangle measure).
We will investigate 3 of the 6 trigonometric
functions:
• tangent
• sine
• cosine
Trigonometry and the Tangent Ratio
Tangent Ratio:
In a right triangle, the ratio of the length of the leg
opposite ‫ﮮ‬P to the length of the leg adjacent to ‫ﮮ‬P
In a right triangle, the ratio of the length of the leg
opposite ‫ﮮ‬P to the length of the leg adjacent to ‫ﮮ‬P .
This is called the tangent ratio.
Tangent of ‫ﮮ‬P = opposite leg
adjacent leg
Write a Tangent Ratio
Write the tangent ratios for ‫ﮮ‬T and ‫ﮮ‬U.
tan T = opp = UV = 3
adj
TV
4
tan U = opp = TV = 4
adj
UV 3
what is a tangent ratio?
•We use tangent ratios to determine side lengths
and angles in right triangles.
•In a right triangle, the ratio of the length of the
leg opposite to an angle to the length of the leg
adjacent to the same angle.
•This ratio is always constant
The Tangent Ratio
A
Tangent of <A:
Length of the leg opposite <A
Length of the leg adjacent to <A
Opp
Tan =
Adj
B
C
Writing Tangent Ratios
Write the tangent ratios for <K and <J:
J
tan<K =
7/3
tan<J =
3/7
7
K
3
L
Find the Tangent Ratios
Find the tangent ratios for <A and <B:
A
tan <A = 2/1 or 2
1
tan <B = 1/2
C
2
B
Find the Tangent Ratios
Find the tangent ratios for <A and <B:
A
tan <A = 6/3 or 2
3
tan <B = 3/6 or 1/2
C
6
B
Solve for the missing side
Find the value of w to the nearest tenth:
10
54°
w
Start at 54°. We have sides opposite and
adjacent of that angle. We can use tangent
to solve for the missing side.
Set up the tangent ratio:
tan 54 = w
10
w = (tan 54)(10)
w = 13.8
Cross multiply
Solve for the missing side
Find the value of w to the nearest tenth:
Start at 57°. We have sides opposite and
adjacent of that angle. We will use tangent
to solve for the missing side.
w
Set up the tangent ratio:
57° 2.5
tan 57 = w
2.5
w = (tan 57)(2.5)
w = 3.8
Cross multiply
Solve for the missing side
Find the value of w to the nearest tenth:
1
w
Start at 28°. We have sides opposite and
adjacent of that angle. We will use tangent
to solve for the missing side.
28°
Set up the tangent ratio:
tan 28 = 1
w
(tan 28)(w) = 1
w=
1
tan 28
Cross multiply
Divide by tan 28
w = 1.9
CHAPTER 7
RIGHT TRIANGLE TRIGONOMETRY
Solving for Angle Measures
The Tangent Ratio
Using the Inverse of Tangent
The Inverse Tangent Button on your
calculator looks like this:
tan-1
(You must press the “2nd” or “SHIFT” or
“INV” Button and then press “tan”)
*We use inverse tangent when solving for a missing
angle measure
Using Inverse Tangent
Find the m < X to the nearest degree:
We will use tangent to find the measure of <X
H
Set up the tangent ratio:
tan X = 5
8
Use inverse tangent:
5
B
8
X
X = tan-1 5
8
X = 32°
If no calculator: divide 5/8, =0.6250
Look under the Tangent column of the trig table for the number closest
to 0.6250, then move your finger to the left to find the degrees (32)
under the Angle column.
Using Inverse Tangent
Find the m < Y to the nearest degree:
Y
We will use tangent to find the measure of <Y
Set up the tangent ratio:
tan Y = 23
25
Use inverse tangent:
25
T
23
P
Y = tan-1 23
25
Y = 43°