solving right triangles
DESCRIPTION
Solving Right Triangles. How do you solve right triangles?. M2 Unit 2: Day 6. To solve a right triangle you need…. 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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/1.jpg)
Solving Right Triangles
M2 Unit 2: Day 6
How do you solve right triangles?
![Page 2: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/2.jpg)
To solve a right triangle you need…..
1 side length and 1 acute angle measure-or-
2 side lengths
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.
![Page 3: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/3.jpg)
Given one acute angle and one side:
•To find the missing acute angle, use the Triangle Sum Theorem.
•To find one missing side length, write an equation using a trig function.
•To find the other side, use another trig function or the Pythagorean Theorem
![Page 4: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/4.jpg)
GUIDED PRACTICE
Example 1
Find m∠ B by using the Triangle Sum Theorem.
180o = 90o + 42o + m∠ B
48o = m∠ B
A
C B48o
7042o
Solve the right triangle. Round decimal answers to the nearest tenth.
Approximate BC by using a tangent ratio.
tan 42o =BC 70
70 tan 42o = BC
70 0.9004 BC
63.0 ≈ BC
Approximate AB by using a cosine ratio.
cos 42o =70 AB
AB cos 42o = 70
AB 70 cos 42o=
AB 70 0.7431
AB 94.2
ANSWER
The angle measures are 42o, 48o, and 90o. The side lengths are 70 feet, about 63.0 feet, and about 94.2 feet.
![Page 5: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/5.jpg)
GUIDED PRACTICESolve a right triangle that has a 40o angle and a 20
inch hypotenuse.
Example 2
Find m∠ X by using the
Triangle Sum Theorem.
180o = 90o + 40o + m∠ X50o = m∠ X
X
YZ
Approximate YZ by using a sine ratio.
sin 40o =XY
2020 ● sin 40o = XY
20 ● 0.6428 ≈
XY
12.9 ≈ BC
Approximate AB by using a cosine ratio.
cos 40o =YZ 20
20 ● cos 40o = YZ20 ● 0.7660 ≈ YZ
15.3 ≈ YZ
The angle measures are 40o, 50o, and 90o. The side lengths are 12.9 in., about 15.3 in., and 20 in.
ANSWER
40o
20 in50o
![Page 6: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/6.jpg)
Solve the right triangle. Round to the nearest tenth.
53
90
30
m Q
m R
m P
PQ
PR
QR
cos53
30
p sin5330
q
37°
18.1
24.0
Example 3
18.1p 24.0q
![Page 7: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/7.jpg)
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.
![Page 8: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/8.jpg)
Calculating Angle Measures from
Trigonometric Ratios
Use your calculator to find each angle measure to the nearest tenth of a degree.
A. cos-1(0.87) B. sin-1(0.85) C. tan-1(0.71)
cos-1(0.87) 29.5° sin-1(0.85) 58.2° tan-1(0.71) 35.4°
Example 4
![Page 9: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/9.jpg)
Inverse trig functions:
Ex: Use a calculator to approximate the measure of the acute angle. Round to the
nearest tenth.
26.6° 20.5° 50.2°
1tan (0.5)m A 1sin (0.35)m A 1cos (0.64)m A
1. tan A = 0.5 2. sin A = 0.35 3. cos A = 0.64
![Page 10: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/10.jpg)
EXAMPLE 2 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 –1 0.87 ≈ 60.5o
b. m ∠ B = cos –1 0.15≈ 81.4o
Example 5
![Page 11: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/11.jpg)
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,
Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°.
RT2 = RS2 + ST2
(5.7)2 = 52 + ST2
Method 2:
Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°.
, so ST = 5.7 sinR.
Example 6
![Page 12: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/12.jpg)
Solve the right triangle. Round decimals the nearest tenth.
3
2
90
AB
BC
AC
m A
m B
m C
Use Pythagorean Theorem to find c…
2 2 32 3
3.6
c
c
3.6Use an inverse trig function to find a missing acute angle…
1 3tan ( ) 56.3
2m A
Use Triangle Sum Theorem to find the other acute angle…
90 56.3 33.7m B
56.3°
33.7°
Example 7
![Page 13: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/13.jpg)
2 2 211 18
21.9
PN
PN
1 11tan ( ) 31.4
18m N
90 31.4 58.6m P
Example 8
![Page 14: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/14.jpg)
2 2 223 7
21.9
TU
TU
1 7cos ( ) 72.3
23m S
90 72.3 17.7m U
Example 9
![Page 15: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/15.jpg)
Solve the right triangle. Round decimals to the nearest tenth.
90 37 53m P
sin3722
13.2
PQ
PQ
cos3722
17.6
QR
QR
90 24 66m T
tan243314.7
TR
TR
33cos24
36.1AT
AT
![Page 16: Solving Right Triangles](https://reader035.vdocuments.mx/reader035/viewer/2022081415/56815a95550346895dc81475/html5/thumbnails/16.jpg)
Homework:Pg 174 (#4-22 even)