Right Triangle Trig, II

Right Triangle Trig, II
Objectives:
1. To define and use
the inverses of sine,
cosine, and tangent
2. To convert between
DMS and decimal
degrees and use
DMS in angle
calculations
Assignment:
• Practice Worksheet
You will be able to define and use
the inverse of sine, cosine, and
tangent
Example 1
What if you knew two sides of
a right triangle, how could
you find the measure of an
angle opposite one of
those sides?
In other words, if the legs of a
right triangle are 3 and 4,
what is the measure of the
angle opposite the smallest
side?
3

4
3
tan  
4
Example 2
What does an inverse do to a function
algebraically and graphically?
Inverses switch inputs and outputs
Inverses reflect a graph over y = x
Inverses give you a way to find the input when you know the output.
Example 2
Explain what x and y represent in y = sin x,
then explain what is meant by the inverse
of y = sin x.
x  input  angle measure
y  output  ratio of sides
The inverse would switch
these:
• Input = ratio of sides
• Output = angle measure
Example 3
For what value of x
is (x, 3) on the
graph of
f(x) = 2x – 5?
Example 4
For what value of x
is (x, 4) on the
graph of
g(x) = x2 – 5?
One-to-One Functions
The difference between the two previous
examples is that the linear function is
one-to-one but the quadratic is not.
In other words, for the linear function every
input has one output, AND every output
has one input.
For functions that are not one-to-one, a given
output may have multiple inputs.
Example 5
Use the graph of f(x) = sin(x) to explain why
a given output may have different inputs.
Inverse Trigonometry
To find an angle measurement in a right
triangle given any two sides, use the
inverse of the trig ratio, but each of them
are only defined on certain intervals.
Inverse Trigonometry
a
1  a 
If sin x  , then x  sin  .
b
b
a
1  a 
If cos x  , then x  cos  .
b
b
a
1  a 
If tan x  , then x  tan  .
b
b
a
90  x  90; 1   1
b
a
0  x  180; 1   1
b
a
90  x  90; 
b
Inverse Trigonometry
a
1  a 
If sin x  , then x  sin  .
b
b
a
1  a 
If cos x  , then x  cos  .
b
b
a
1  a 
If tan x  , then x  tan  .
b
b
• “sin-1 x” is read
“the angle
whose sine is x”
or “inverse sine
of x”
• arcsin x is the
same thing as
sin-1 x
Example 5
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.
1. sin A = 0.87
2. cos B = 0.15
Type in your calculator as:
sin 1 (0.87)
Example 6
If the legs of a right triangle
are 3 and 4, what is the
measure of the angle
opposite the smallest side?
3

4
Example 7
Find the measures of the
acute angles of a 8-15-17
right triangle.
Example 8
Suppose your school is building a raked stage. The
stage will be 30 feet long from front to back, with a
total rise of 2 feet. A rake (angle of elevation) of 5°
or less is generally preferred for the safety and
comfort of the actors. Is the raked stage you are
building within this suggested range?
Example 9
To solve a right triangle means to find all of
its sides and angles. Using trigonometry,
what must you know to solve a right
triangle?
Example 10
Solve the right triangle.
Round your answers
to the nearest tenth.
42
70 ft
Example 11
Solve each right triangle. Write your
answers in simplest radical form.
1 unit
1 unit
30
45
Objective 2
You will be able to convert between
DMS and decimal degrees and use
DMS in angle calculations
Degrees Are Too Big
Sometimes a degree is just too
big. In that case, we can
break a degree into 60
equal parts, called minutes
(‘). For those of you who
think a minute is too big, we
can break each of those into
60 equal parts, called
seconds (“). We’ll call
angle measures in degrees,
minutes, and seconds DMS.
DMS
Based on the previous
divisions:
1 degree = 60 minutes
1 minute = 60 seconds
33°12’14” reads 33
degrees, 12 minutes, and
14 seconds
DMS
Based on the previous
divisions:
1° = 60’
1’ = 60”
33°12’14” reads 33
degrees, 12 minutes, and
14 seconds
Exercise 12
If there are 60 minutes in a degree and 60
seconds in a minute, how many seconds
are there in a degree?
1 degree = 3600 seconds
1° = 3600”
DMS to Decimal Degrees
Although accurate, DMS are not always
convenient for calculations. In this case,
we’d want to convert DMS to decimal
degrees. To do this, just write your DMS
measure as the sum of fractions.
12   14 

3312'14"  33      

 60   3600 
 33.2038
Exercise 13
Convert from DMS to decimal degrees.
1. 97°4’35”
2. 13°32’50”
Decimal Degrees to DMS
On the other hand, sometimes we would want to
convert decimal degrees into DMS.
32.159  329'32.4"
D
Take off the whole number. This is the
number of degrees.
32°
M
Multiply the remaining decimal by 60.
Take off the whole number. This is the
number of minutes
(.159)(60) = 9.54
9’
S
Multiply the remaining decimal by 60.
This is the number of seconds.
(.54)(60)=32.4
32.4”
Exercise 14
Convert decimal degrees to DMS.
1. 20.5°
2. 46.327°
3. -189.62°
Exercise 15
Let θ = 56°34’53”. Find the complement and
supplement to θ.
Right Triangle Trig, II
Objectives:
1. To define and use
the inverses of sine,
cosine, and tangent
2. To convert between
DMS and decimal
degrees and use
DMS in angle
calculations
Assignment:
• Practice Worksheet