Post on 23-Dec-2015
60º5
?
45º8
?
Recall: How do we find “?”
65º5
?
What about this one?
60º5
?
What is the ratio of long leg to short leg?
3
1
60º11
?
60º7
?
65º5
?
65º12
?
65º123
?
These triangles are all similar (AA~).
What is the relationship of their ratios of long leg to short leg?
The ratios are all the same.
Right Triangle Trigonometry
Sections 9.1 and 9.2
What is Trigonometry?
x 70
20
x
3020
Angle ProblemTriangle Sum Theorem
Side ProblemPythagorean Theorem
x
30
20
Angle and Side Problem
Tangent Ratio
Opposite Leg of
Adjacent Leg
BCTan A
AC
AdjacentLeg
OppositeLeg
B
CAAdjacentLeg
OppositeLeg
B
CA
Opposite Leg of
Adjacent Leg
ACTan B
BC
Trig ratios are always with respect to a specific angle.
Labeling in a right triangle
a
b
B
CA
c
tan A2021
= = =oppositeadjacent
BCAC
tan B2120
= = =oppositeadjacent
ACBC
Write the tangent ratios for A and B.
Calculator Trig Functions
37°
B
CA
(37 )Tan
Make sure the calculator is set to “degrees”
Opposite
Adjacent Angle Measure
0.7536
If you must round, use at least 3 decimal places.
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.
tan 32° = height125 Use the tangent ratio.
height = 125 (tan 32°) Solve for height.
125 32 78.108669 Use a calculator.
The tree is about 78 ft tall.
Sine Ratio
Opposite Leg of
Hypotenuse
BCSin A
AB
Opposite Leg of
Hypotenuse
ACSin B
AB
OppositeLeg
Hypotenuse
B
CA
OppositeLeg
Hypotenuse
B
CA
Cosine Ratio
Adjacent Leg of
Hypotenuse
ACCos A
AB
AdjacentLeg
Hypotenuse
B
CA
AdjacentLeg
Hypotenuse
B
CA
Adjacent Leg of
Hypotenuse
BCCos B
AB
Use the triangle to find sin T, cos T, sin G, and cos G. Write your answer in simplest terms.
sin T = =1220
35=
oppositehypotenuse
cos T = =1620
45=
adjacenthypotenuse
sin G = =1620
45=
opposite hypotenuse
cos G = =1220
35=
adjacent hypotenuse
Calculator Trig Functions
37°
B
CA
(37 ) 0.6018Sin
(37 ) 0.7986Cos
(37 ) 0.7536Tan
Make sure the calculator is set to “degrees”
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.
cos 35° =height
20 Use the cosine ratio.
height = 20 • cos 35° Solve for height.
20 35 16.383041 Use a calculator.
The flagpole is about 16 ft tall.
SOH-CAH-TOA
Opposite Leg of
HypotenuseSin
Adjacent Leg of
HypotenuseCos
Opposite Leg of
Adjacent LegTan
SOH
CAH
TOA
SOH-CAH-TOA
Inverse Trig Functions
x°
B
CA
If the Sin of an angle is 0.8191, what is the measure of the angle?
1(0.8192)Sin
Opposite
Hypotensue Angle Measure
55
Regular vs. Inverse
(Angle measure)Opposite
TanAdjacent
1 Angle measureOpposite
TanAdjacent
(Angle measure)Opposite
SinHypotenuse
1 Angle measureOpposite
SinHypotenuse
(Angle measure)Adjacent
CosHypotenuse
1 Angle measureAdjacent
CosHypotenuse
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.
cos A =1.54.0 0.375= Use the cosine ratio.
Use the inverse of the cosine.m A = cos–1(0.375)
Use a calculator.0.375 67.975687
Round to the nearest degree.m A 68
(continued)
To find m B, use the fact that the acute angles of a right triangle are complementary.
The acute angles, rounded to the nearest degree, measure 68 and 22.
m A + m B = 90 Definition of complementary angles
Substitute.68 + m B 90
m B 22
Find m R to the nearest degree.
tan R =4741 Find the tangent ratio.
So m R 49.
m R tan–1 Use the inverse of the tangent.4741
Use a calculator.48.9004944741