math-2 lesson 5-10 -...
TRANSCRIPT
Math-2Lesson 5-10
Solving Right Triangles
using Trig Ratios
Trigonometric Ratios
h
osin A
Key point: we MUST write
“code word” and “reference angle”
h
acos A
a
otan A
These only work for right triangles!!!
2
1
h
o30sin
In order to get the associated ratio.
Practice: Only side lengths have been provided.
5
12
13
A
B
C
What is the tangent ratio for angle B?
What is the sine ratio for angle A?
1
2 Y
X
Z
What is the cosine ratio for angle X?
What is the sine ratio for angle Y?
5
13
5sin A
5
5
5
1cos X
5
12tan B
5
5
5
1sin Y
?tan B
?sin A
?cos X
?sin Y
Angle hyp
oppA sin
0.173610
0.342020
0.530
0.69349.430.866060
If we know the numerical value of the reference angle,
either a:
(1) Trigonometric Table
(2) calculator
can determine the ratio associated with that angle.
Ratios are not given in “fraction form”
0.174
AC
B
Z Y
X
D=10°
A=10°
174
hyp
opp:ratio sine 174.010sin
1
1
174.0174.010sin
Ratios are not given in “fraction form”
But they come from a triangle.
hyp
oppA sine
To find an unknown value we need an equation.
What equation do we have that is a relationship between
just the measures of ANGLES ?
(1) Triangle Angle Sum Theorem:
measures of angles add up to 180° 180 CmBmAm
What equation do we have that is a relationship between
just the measures of the SIDES ?
(2) Pythagorean Theorem 222 cba
(3) Trig Ratios
What equations do we have that is are relationships
between the measures of BOTH the SIDES and ANGLES?
h
osin A
h
acos A
a
otan A
Solve a triangle: finding the unknown
measures of angles and sides. 6
b
7
A
Solving this triangle did not
require trigonometry. BUT we
could have used it for the sides.
90 BmAm
B
222 76 b
13b
59º
C
9059 Bm
31Bm222 cba
4936 2 b
132 b
1) Angles
2) Sidesa
otan A
6
b31tan
h
acos A
7
b59cos
7
b515.0
6.3)7(515.0 b6.3b
7
b601.0
6.3)6(601.0 b
2) Sides (must use trigonometry)
Solve the Triangle
y
x
12
A90 BmAm
B
43º
C9043 Bm
47Bm
1243sin
y
47º
y)682.0(12
y=8.2
1243cos
x
12731.0
x
8.8x
x=8.8
h
oA sin
1) Angles
12682.0
y
2.8y
h
aA cos
x)731.0(12
Solve the Triangle
y
x
32
A90 BmAm
B
25º
C9025 Bm65Bm
3225sin
y
65ºy=13.5
3225cos
x
x = 29
h
oA sin
2) Sides (must use trigonometry)
1) Angles
y)423.0(32
32423.0
y
5.13y
32906.0
x
29x
h
aA cos
x)906.0(32
43ºAC
B
47º
5
If you don’t know what Trig. Ratio equation to use
write equations for all of them (to see which ones
only have one unknown value
bsin 43º = 5/c
c = 7.3
c
tan 47º = b/5
b = 5.4
cos 43º = b/c
One equation, two unknowns
infinitely many
combinations of x-y values.
c = 7.3
0.682= 5/c
0.682c= 5
c= 5/0.682
1.072= b/5
b = 5(1.702)
b = 5.4
Your turn: Solve the triangle
1.
2.
3.
y
x
12
A37º
C
53º?Bm
B
?x
?y
Your turn: Solve the triangle
4.
5.
6.
10
x
r
A51º
C
39º?Bm
B
?x
?r
12ºA
B
C
c
Angle C is a right angle. Solve the
triangle.
78Bma
2.3
b = 2.312Am
1) Draw the standard triangle and label it.
2) Solve the triangle.
c
a25sin
3.212tan
a
a
oA tanh
aA cos
One equation, two unknowns
infinitely many
combinations of x-y values.
c
3.212cos
3.2213.0
a
c
3.2978.0
978.0
3.2c
4.2c
5.0)213.0(3.2 a
A
C
B
Given the following information about a right triangle, solve
the triangle: ‘a’ = 5, A = 18