Transcript solve a triangle

SECTION 14-5
• Applications of Right Triangles
Slide 14-5-1
APPLICATIONS OF RIGHT TRIANGLES
•
•
•
•
•
Calculator Approximations for Function Values
Finding Angles Using Inverse Functions
Significant Digits
Solving Triangles
Applications
Slide 14-5-2
CALCULATOR APPROXIMATIONS FOR
FUNCTION VALUES
Because calculators differ among makes and models,
students should always consult their owner’s manual
for specific information concerning their use.
When evaluating trigonometric functions of angles
given in degrees, it is a common error to use the
incorrect mode; remember that the calculator must
be set in the degree (not radian) mode.
Slide 14-5-3
EXAMPLE: FINDING FUNCTION VALUES
WITH A CALCULATOR
Use a calculator to approximate the value of each
trigonometric function.
a) sin 5412
b) sec32.662 c) tan(321)
Solution
a) sin 5412  sin(54.20)  .81106382
1
b) sec32.662 
 1.18783341
cos 32.662
c) tan(321)  .80978403
Slide 14-5-4
FINDING ANGLES USING INVERSE
FUNCTIONS
We have used a calculator to find trigonometric
function values of angles. This process can be reversed
using inverse functions. Inverse functions are denoted
by using –1 as a superscript. For example, the inverse
of f is denoted f –1. For now we restrict our attention
to angles  in the interval 0    90. The measure
of an angle can be found from one of its trigonometric
function values using inverse functions as show on the
next slide.
Slide 14-5-5
EXAMPLE: USING INVERSE FUNCTIONS
TO FIND ANGLES
Use a calculator to find a value for  such that
0    90. and  satisfies each of the following.
a) sin   .715332
b) sec   1.21112
Solution
Use sin-1 (2nd and then sin).
a)   45.6704
1
1
b) cos 

 .8256820133
sec  1.21112
  34.3423
Use cos-1.
Slide 14-5-6
SIGNIFICANT DIGITS
A significant digit is a digit obtained by actual
measurement. A number that represents the result of
counting, a a number that results from theoretical work
and is not the result of measurement, is an exact
number.
Most values of trigonometric functions are
approximations, and virtually all measurements are
approximations. To perform calculations on such
approximate numbers, follow the rules on the next
slide.
Slide 14-5-7
CALCULATION WITH SIGNIFICANT
DIGITS
For adding and subtracting, round the answer so that
the last digit you keep is in the rightmost column in
which all the numbers have significant digits.
For multiplying or dividing, round the answer to the
least number of significant digits found in any of the
given numbers.
For powers and roots, round the answer so that it has
the same number of significant digits as the numbers
whose power or root you are finding.
Slide 14-5-8
SIGNIFICANT DIGITS FOR ANGLES
Number of
Significant Digits
2
3
4
5
Angle Measure to the Nearest:
Degree
Ten minutes, or nearest tenth of a
degree
Minute, or nearest hundredth of a
degree
Tenth of a minute, or nearest
thousandth of a degree
Slide 14-5-9
SOLVING TRIANGLES
To solve a triangle means to find the measures of all
the angles and all the sides of a triangle. In using
trigonometry to solve triangles, a labeled sketch is
useful.
B
c
a
C
b
A
Slide 14-5-10
EXAMPLE: SOLVING A RIGHT TRIANGLE
GIVEN AN ANGLE AND A SIDE
Solve the triangle below.
Solution
a
sin 40.3 
14.8
a  (sin 40.3)14.8  9.57
b
cos 40.3 
14.8
b  (cos 40.3)14.8  11.3
B
14.8
a
C
40.3 A
b
B = 90° – A
= 90° – 40.3° = 49.7°
Slide 14-5-11
EXAMPLE: SOLVING A RIGHT TRIANGLE
GIVEN TWO SIDES
Solve the triangle below.
B
Solution
a
23.1
cos A 
42.5
C
A  57.1
42.5
23.1
A
B = 90° – 57.1° = 32.9°
a b  c
2
2
2
a  23.1  42.5
a  35.7
2
2
2
Slide 14-5-12
APPLICATIONS
The angle of elevation from point X to point Y (above
X) is the acute angle formed by ray XY and a horizontal
ray with endpoint at X. The angle of elevation is
always measured from the horizontal.
The angle of depression from point X to point Y
(below X) is the acute angle formed by ray XY and a
horizontal ray with endpoint at X.
Y
X Horizontal
Angle of
elevation
X
Horizontal
Y
Angle of
depression
Slide 14-5-13
EXAMPLE: ANGLE OF ELEVATION
The length of the shadow of a building 35.28 meters
tall is 40.45 meters. Find the angle of elevation of the
sun.
Solution
35.28
tan  
40.45
  41.09
35.28 m
The angle of elevation of
the sun is 41.09°.
40.45 m
Slide 14-5-14