Download Trigonometry

Unit Circle
The values for sine and cosine create a
nice pattern.
If we let cos θ be the x value and sin θ be
the y value, the plot looks like an arc.
There are some values of for which we
can find the exact values of cos θ and sin
θ; these are the values found on the
special right triangles.
Special Right Triangles
s√2
s
s
2s
s√3
s
The Unit Circle: Quadrant I
Trigonometric Functions: θ>90°
Even though we can’t construct a right
triangle with non-acute angles, we can find
the sine and cosine of angles larger than
90°.
These values make sense for application
problems that we will look at later.
To find sin θ, cos θ, etc., when θ>90°, we
need to calculate θ’s reference angle.
Reference Angle
The reference angle of θ is the acute
angle formed by the terminal ray of θ and
the x-axis.
Ex.: θ=132°, θref=48° ; θ=275°, θref=85°
Reference angles allow us to construct the
remainder of the circle.
The Unit Circle: Quadrants II–IV
The trigonometric functions of θ are the
same magnitude (numerical size) as those
of θref.
The difference may be the size, which is
determined by the quadrant in which the
angle lies.
Sine and Cosine
Because sine is the y value on the unit
circle, it is positive for angles in quadrants
I and II and negative for angles in
quadrants III and IV.
Because cosine is the x value on the unit
circle, it is positive for angles in quadrants
I and IV and negative for angles in
quadrants II and III.
The Other Trigonometric Functions
Once we know sine and cosine, we can
find the other trigonometric functions using
identities.
An identity is a mathematical statement
that is always true (as opposed to an
equation, which is only true for some
values).
The Six Trigonometric Functions
Sine
Cosecant
opp
sin  
hyp
1
hyp
csc  

sin  opp
Cosine
Secant
adj
cos  
hyp
1
hyp
sec  

cos  adj
Tangent
Cotangent
sin  opp
tan  

cos  adj
1
cos  adj
cot  


tan  sin  opp
Positive or Negative?
The mnemonic “All Students Take
Calculus” can be used to determine in
which quadrants sine, cosine, and tangent
are positive.
I: All
II: Sine +, Cosine and Tangent –
III: Tangent + , Sine and Cosine –
IV: Cosine –, Sine and Tangent +

(Note: reciprocals always have the same sign... so secant is positive wherever cosine is, etc.)
You Should Be Able To…
Use the unit circle to find the sine and
cosine for the angles that come from the
special right triangles.
Determine reference angles and use the
unit circle to find the sine and cosine for
non-acute angles.
Find the other functions (tangent, etc.)
based on the ratios of sine and cosine.
Quick Check
Find the reference angle and the exact
values of the six trigonometric functions of:
θ=7π/6.