Transcript Trigonometric Ratios

Warm-Up 1
Find the value of x.
History Lesson
Right triangle
trigonometry is the
study of the relationship
between the sides and
angles of right triangles.
These relationships can
be used to make
indirect measurements
like those using similar
triangles.
History Lesson
Early mathematicians
discovered trig by measuring
the ratios of the sides of
different right triangles. They
noticed that when the ratio of
the shorter leg to the longer
leg was close to a specific
number, then the angle
opposite the shorter leg was
close to a specific number.
Example 1
In every right triangle in which the ratio of the
shorter leg to the longer leg is 3/5, the
angle opposite the shorter leg measures
close to 31. What is a good
approximation for x?
Example 2
In every right triangle in which the ratio of the
shorter leg to the longer leg is 9/10, the
angle opposite the shorter leg measures
close to 42. What is a good
approximation for y?
Trig Ratios
The previous examples worked because the
triangles were similar since the angles
were congruent. This means that the
ratios of the sides are equal.
In those cases we were using the tangent
ratio. Here’s a list of the three you’ll have
to know.
sine
cosine
tangent
Trigonometric Ratios I
Objectives:
1. To discover the three main trigonometric
ratios
2. To use trig ratios to find the lengths of
sides of right triangles
Summary

A
side adjacent Θ
B
side opposite Θ
C
sin  opposite
hypotenuse
cos  adjacent
hypotenuse
tan  opposite
adjacent
Summary

A
side adjacent Θ
B
side opposite Θ
C
Oh
sin  Oh
Hell
Heck
cos  Another
Hour
tan 
Of
Algebra
SohCahToa
Soh
sin  opposite
hypotenuse
Cah
cos  adjacent
hypotenuse
Toa
tan  opposite
adjacent
Example 3
Find the values of the six trig ratios for α and
β.
Activity: Trig Table
Step 5: Finally, let’s check your values with
those from the calculator.
For sin, cos, and tan
1. Make sure your calculator is set to DEGREE in the
MODE menu.
2. Use one of the 3 trig keys. Get in the habit of
closing the parenthesis.
Example 4
To the nearest meter,
find the height of a
right triangle if one
acute angle
measures 35° and
the adjacent side
measures 24 m.
Example 5
To the nearest foot, find the length of the
hypotenuse of a right triangle if one of the
acute angles measures 20° and the
opposite side measures 410 feet.
Example 8
Find the value of x to the nearest tenth.
1. x =
2. x =
3. x =