Transcript Exact Values

Computing the Values of Trig
Functions of Acute Angles
TRIANGLES
The 45-45-90 Triangle
In a 45-45-90 triangle the sides are in a ratio of 1- 1- 2
This means I can build a triangle with these lengths for sides
(or any multiple of these lengths)
We can then find the six trig
functions of 45° using this triangle.
sin 45 
cos45 
0 1
t an 45    1
a 1
rationalized
45°
2
1
45°
Can "flip" these to
get other 3 trig
functions
90°
1
You are expected to know exact values for trig functions of 45°.
You can get them by drawing the triangle and using sides.
What is the radian equivalent of 45°?

4
You also know all the trig
functions for /4 then.
45°

2
sec 
 2
1
4
reciprocal of cos so h over a

1
t an   1
4 1
2
1
45°
90°
1
The 30-60-90 Triangle side opp 60°
In a 30-60-90 triangle the sides are in a ratio of 1- 3 - 2
side opp 90°
side opp 30°
This means I can build a triangle with these lengths for sides
We can then find the six trig functions of 30°or 60°
using this triangle.
I used the triangle and
o 1
sin 30  
h 2
a 1
cos60  
h 2
t an30 
did adjacent over
hypotenuse of the 60°
to get this but it is the
cofunction of sine so
this shows again that
cofunctions of
complementary angles
are equal.
30°
2
3
60°
90°
1
Be sure to locate the angle you want
before you find opposite or adjacent
What this means is that if you memorize the special triangles,
then you can find all of the trig functions of 45°, 30°, and 60°
which are common ones you need to know.
You also can find the radian equivalents of these angles.
45  

4
30  

6
60  

3
When directions say "Find the exact value", you must
know these values not a decimal approximation that
your calculator gives you.
Using a Calculator to Find
Values of Trig Functions
If we wanted sin 38° we could not use the
previous methods to find it because we
don't know the lengths of sides of a
triangle with a 38° angle. We will then
use our calculator to approximate the
value.
You can simply use the sin button on
the calculator followed by (38) to find
the sin 38°
A word to the wise: Always make sure your calculator
is in the right mode for the type of angle you have
(degrees or radians)
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au