Transcript Solving Right Triangles - Effingham County Schools

MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
How do you solve right triangles?
Lesson 5.4
Day 53
Saturday, April 2, 2016
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Every right triangle has one right angle, two acute
angles, one hypotenuse, and two legs.
To SOLVE A RIGHT TRIANGLE means to find
all 6 parts.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
1. Find the measure of the missing angle.
Round your answer to the nearest degree.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
2. Find the measure of the missing angle.
Round your answer to the nearest degree
.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
3.
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Find the missing side. Round your answer to the nearest tenth.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
4.
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Find the missing side. Round your answer to the nearest tenth.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
5.
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Solve the triangle. Round your answers to the nearest tenth.
R
RT  14.9
S
T
mR  20.7 o
mS  90o
mT  70.3o
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
6.
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Solve the triangle. Round your answers to the nearest tenth.
N
O
M
MO  21.6
mN  26.1o
mM  90o
mO  63.9o
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
Y
125
Z
z
50
X
YX » 134.6
mY  21.8
mX  68.2
8. Solve the right triangle. Round decimals to the tenth.
(Hint find all missing side lengths and angle measures)
P
22
37˚
Q
PQ  13.2
R
QR  17.6
mQ  90
mP  53
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
Hint: If you know any
trig ratio, use your
calculator to find the
missing angle
Y
5
X
13
12
Z
12
sinY 
13
 12 
sin1 

13


mY  67.4
mZ  90  67.4
mZ  22.6
2. Solve the right triangle. Round decimals to the tenth.
(Hint find all missing side lengths and angle measures)
2
a +b =c
A
2
17
C
cos A 
15
17
2
2
2
a + 15 = 17
2
a + 225 = 289
2
a = 64
a= 8
15
B
2
 15 
cos 

 17 
mA  28.1
mB  90  28.1
mB  61.9
1
MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles.
GUIDED PRACTICE
Solve the right triangle. Round decimal answers to the nearest tenth.
A
Example 5
42o
Find m∠ B by using the Triangle Sum Theorem.
c
180o = 90o + 42o + m∠ B
70
48o = m∠ B
Approximate BC by using a tangent ratio.
a
tan 42o =
70
63.0 ≈
a
Approximate AB by using a cosine ratio.
70
cos 42o =
c
94.2
c
48o
C
a
B
ANSWER
The angle measures are
42o, 48o, and 90o. The
side lengths are 70 feet,
about 63.0 feet, and
about 94.2 feet.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles.
GUIDED PRACTICE
Solve a right triangle that has a 40o angle and a 20 inch hypotenuse.
Example 6
X
Find m∠ X by using the Triangle Sum Theorem.
180o
=
90o
+
40o
50o
+ m∠ X
20 in
50o = m∠ X
Approximate YZ by using a sine ratio.
XY
sin 40o =
20
12.9 ≈ XY
Approximate YZ using a cosine ratio.
YZ
cos 40o =
20
15.3 ≈
YZ
40o
Y
Z
ANSWER
The angle measures are 40o, 50o,
and 90o. The side lengths are 12.9
in., about 15.3 in., and 20 in.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Example 7
Solve the right triangle. Round to the nearest tenth.
mQ  53
mR  90
mP  37°
PQ  30
PR  24.0
p
cos53 
30
p  18.1
q
30
q  24.0
sin53 
QR  18.1
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
If you know the sine, cosine, or tangent of an acute angle measure,
you can use the inverse trigonometric functions to find the measure of
the angle.
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2 Students will define and apply sine, cosine, and tangent ratio to right triangles.
EXAMPLE 2
Example 8
Use an inverse sine and an inverse cosine
Let ∠ A and ∠ B be acute angles in a right triangle. Use a calculator to
approximate the measures of ∠ A and ∠ B to the nearest tenth of a
degree.
a.
sin A = 0.87
b.
cos B = 0.15
SOLUTION
a.
m∠
A
= sin
b.
m∠
B
= cos
–1
–1
0.87 ≈ 60.5o
o
0.15≈ 81.4
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Example 6
Solving Right Triangles
Find the unknown measures. Round lengths to
the nearest hundredth and angle measures to
the nearest degree.
Method 1: By the Pythagorean Theorem,
Method 2:
RT2 = RS2 + ST2
(5.7)2 = 52 + ST2
Since the acute angles of a right
triangle are complementary, mT 
90° – 29°  61°.
,
Since the acute angles of a right
triangle are complementary,
mT  90° – 29°  61°.
ST
sin 29 
5.7
o
ST  5.7 sin 29
ST  2.76
o
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Example 7
Solve the right triangle. Round decimals
the nearest tenth.
Use Pythagorean Theorem to find c…
c 2  22  3 2
c  3.6
Use an inverse trig function to
find a missing acute angle…
3
mA  tan ( )  56.3
2
1
Use Triangle Sum Theorem to
find the other acute angle…
mB  90  56.3  33.7
AB  3.6
BC  3
AC  2
mA  56.3°
mB  33.7°
mC  90
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Example 8
PN 2  112  18 2
PN  21.1
11
mN  tan ( )  31.4
18
1
mP  90  31.4  58.6
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Example 9
232  TU 2  72
TU  21.9
7
mS  cos ( )  72.3
23
1
mU  90  72.3  17.7
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
tan 55o 
55o
555
g
g  388.6 ft.
555
555 ft
55o
g
55o
g
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
 Homework:
◦ Pg 174 (#4-22 even)
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.
MM2G2
Students will define and apply sine, cosine, and tangent ratio to right triangles.
Ladder Problems
http://www.geogebra.org/en/examples/ladder_wall/ladder_wall.html
MM2G2b Explain the relationship of the trigonometric ratios of complementary angles.
MM2G2c Solve application problems using the trigonometric ratios.