Transcript Trigonometric Ratios

Trigonometric Ratios II
Objectives:
1. To find missing angles in a right triangle
using inverse trigonometric ratios
2. To complete and use the unit circle to find
the exact values of various angle
measures
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  , t hen x  sin  .
b
b
a
1  a 
If cos x  , t hen x  cos  .
b
b
a
1  a 
If t an x  , t hen x  t an  .
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  , t hen x  sin  .
b
b
a
1  a 
If cos x  , t hen x  cos  .
b
b
a
1  a 
If t an x  , t hen x  t an  .
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
Radians
Radians
Radians are
another way to
measure an
angle. If you
take the radius
and wrap it
around the circle,
the angle that is
formed is one
radian.
Radians
It takes a little bit
more than 3
radians to span a
semicircle.
That “little bit more
than 3” is π.
So π radians = 180°
and 2π radians =
360°
Example 10
Rewrite each of the following angle
measures in terms of radians.
(180° = π rad)
1. 30°
2. 45°
3. 60°
4. 90°
The Unit Circle
This tiny circle is
called the unit
circle since its
radius is 1 unit.
This circle may be
tiny, but it will give
us a way to find
102 exact trig
values. That’s
pretty useful.