Download 8.3 Solving Right Triangles

8.3 Solving Right Triangles
Objectives: G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
For the Board: You will be able to solve a right triangle.
Anticipatory Set:
Every right triangle has one right angle, two acute angles,
one hypotenuse, and two legs.
hypotenuse
leg
leg
To solve a right triangle means to determine the measures of all six parts.
There are two solvable situations.
I. Given the measure of an acute angle and the measure of any one of the sides of the triangle.
Steps: 1. Since one angle is 90° and a second is given, you can find the remaining angle by using
90° + m<1 + m<2 = 180°.
2. Determine how the known side is related to the known angle. Is it opposite,
adjacent, or hypotenuse? (Remember the hypotenuse is ALWAYS across from the
right angle.)
3. Since there are two unknown sides, pick one of the two to find first. Determine how
this side is related to the known angle. Is it opposite, adjacent, or hypotenuse?
4. Decide which trigonometric function includes everything from above. Set up and
solve the trigonometric function for the unknown side.
5. Now that you know 2 sides of the triangle, you can find the missing side by using the
Pythagorean Theorem (a2 + b2 = c2).
Example: Find x, y, and z, in the given triangle.
Since m<T = 61°, m<R = 90° – 61° = 29°, z° = 29°
9 is opposite <T
Solve for x: x is adjacent to <T
Use tan.
tan 61 = 9/x
x = 9/tan 61 = 4.99
4.992 + 92 = y2
y2 = 105.90
y = 10.29
White Board Activity:
Practice: Find x, y, and z, in the given triangle.
a.
S
x
T
5
z°
56°
R
y
z: 90 – 56 = 34°
x: tan 56° = 5/x
1.48 = 5/x
x = 5/1.48 = 3.38
y: 3.382 + 52 = y2
y = 6.04
b.
S
7
S
x
z°
61°
T
x
z°
65°
9
R
y
T
z: 90 – 65 = 25°
y: cos 65° = 7/y
0.42 = 7/y
y = 7/0.42 = 16.67
x: 72 + x2 = 16.672
x = 15.13
y
R
II. Given the measure of any two sides.
Steps: 1. Use the Pythagorean Theorem to find the remaining side.
2. Since there are two unknown angles, pick one of them.
3. Determine which trigonometric function would involve the chosen angle and the two
GIVEN sides. Do not use the side you found by the Pythagorean Theorem because it
is probably a rounded answer.
4. You must now “undo” the trigonometric function to find the angle. This is done by
using the inverse trigonometric functions.
For an acute angle:
 opposite 

 = Angle.
 hypotenuse
 adjacent 
adjacent
 = Angle.
If cos (Angle) =
, then cos-1 
hypotenuse
 hypotenuse
If sin (Angle) =
opposite
, then sin-1
hypotenuse
If tan (Angle) =
opposite
, then tan-1
adjacent
 opposite

 = Angle.
 adjacent 
There are two types of calculators which can do trigonometric functions:
algebra logic, and non-algebra logic.
Algebra Logic Key Strokes
Sin-1 (INV or 2nd SIN)
Cos-1 (INV or 2nd COS)
Tan-1 (INV or 2nd TAN)
(
ratio
)
=
Non-Algebra Logic Key Strokes
(
Ratio
)
Sin-1 (INV or 2nd SIN)
Cos-1 (INV or 2nd COS)
Tan-1 (INV or 2nd TAN)
Example: Use your calculator to find each angle measure to the nearest degree.
a. cos-1 (0.26)
b. sin-1 (0.55)
c. tan-1 (0.73)
75°
33°
36°
White Board Activity:
Practice: Use your calculator to find each angle measure to the nearest degree.
a. cos-1 (8/17)
b. sin-1 (5/13)
c. tan-1 (5/4)
62°
23°
51°
Open the book to page 553 and read example 3.
S
Example: Find the unknown measures. Round side lengths
15
to the nearest hundredth and angle measures
y°
to the nearest degree.
T
152 + 182 = x2
x2 = 549
x = 23.43
Solve for y: 15 is adjacent and 18 is opposite, so use tan-1.
tan-1 ( 18/15) = 50°
z° = 90° – 50° = 40°
White Board Activity:
Practice: Find the unknown measures. Round side lengths to the nearest
hundredth and angle measures to the nearest degree.
a.
b.
5
14
S
x
z°
y°
R
5.7
T
S
10
z°
y°
18
z°
R
x
R
x
T
5.72 – 52 = x2 cos z = 5/5.7
x = 2.74
cos-1 (5/5.7)= z°
z° = 29°
y° = 90 – 29 = 61°
102 + 142 = x2
x = 17.20
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 555 – 558 prob. 1 – 15, 20 – 35, 38 – 44, 48 – 50, 54 – 57.
For a Grade:
Text: pgs. 555 – 558 prob. 22, 28, 34, 38, 54.
tan z = 10/14
tan-1 (10/14) = z
z° = 35.5°
y° = 90 – 35.5
= 54.5°