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Trigonometry
Right Triangle Trig
Example 1: Find the exact values of sine, cosine, and tangent for angle θ below:
Common Pythagorean Triples:
Common Right Triangles:
45◦ − 45◦ − 90◦
30◦ − 60◦ − 90◦
The Radian
Consider an object moving along a circle:
As the point moves about the circle, there is a direct relationship between the angle θ and the
distance traveled s (Called the arc length).
Consider some cases:
.
Radian measure was defined to make the distance calculations (on the right above) as simple as
possible; to make the arc length formula
s = rθ
Notice - from the pattern above, this means a full rotation corresponds to an angle of
Definition: Radians are a unit of measurement for angles such that 2π radians corresponds to a
full rotation around a circle.
Notice - If the arc length is equal to the radius of the circle (s = rθ becomes r = rθ), then the
measure of the angle is θ = 1 radian. This leads to an equivalent definition:
If an angle θ sweeps out an arc of length equal to the radius r, then the measure of the angle θ is
1 radian.
Remarks:
• When angles are given as unitless numbers, assume they are measure in radians. If an angles
is measure in degrees, you must include the degree symbol.
• MANY formulas/definitions in Calculus are much simpler when we use radians instead of
degrees. Thus, you MUST become comfortable thinking in radians.
Example 4: In which quadrant do the following angles lie?
2π
(a) θ =
3
(b) θ =
7π
6
(c) θ = 1
(d) θ =
15π
8
Angles larger than 2π or less than 0:
Example 5: Sketch each angle in standard position.
5π
(a) θ =
2
(b) θ = −
3π
4
Trig Definitions with the Unit Circle
Let θ be an angle in a circle of radius r. Suppose θ falls in Quadrant I:
Notice the right triangle here - using the right triangle trig definitions, we can see there is a relationship between the trig ratios of θ and the point (x, y) where the angle intersects the circle:
While we do not have time to prove it here, this relationship holds for all angles (not just those
that fall in QI). When using this relationship, the easiest circle to consider would be one with radius
r = 1 - the Unit Circle!
Definition: For any angle θ in standard position and the associated point P (x, y) on the unit circle,
And we can extend the relationship between the point P (x, y) and θ to define the other trig ratios:
Example 6: Suppose sin θ = −
1
and θ lies in QIII. Determine cos θ.
3
While the above definition is true for any angle θ, there are certain common angles that get used
often - hence the picture you may have seen before. (see link on course website)
It is important that you understand both the right triangle and the unit circle definitions and see
how they fit together.
3π
3π
Example 7: What is sin
? What is sec
?
2
2
Example 8:
13π
Evaluate csc −
.
6
Sine and Cosine as Functions of the Real Numbers
Recall - radians relate angles to arc-length: s = rθ.
On the unit circle, r = 1. So s = θ.
This is a subtle but powerful difference!
We can now think of sine and cosine as functions of real numbers (as opposed to ratios of the lengths
of the sides of a triangle or coordinates on a circle). We can also now sketch the graphs of each
function by plotting the points (s, sin(s)) and (s, cos(s)).
1) f (s) := sin(s)
f (x) := sin x.
• Domain
• Range
• Intercepts
2) f (s) := cos(s)
Tangent Function
f (x) := tan x
Domain:
So the domain of f (x) := tan x is all real numbers except x =
What is happening at the ”ends” of this domain?
Consider the value x =
π
:
2
π
+ kπ, k an integer.
2
Similar logic can be used to show that the behavior around each of the other ”ends” of the domain
3π
is the same. For example, for x =
:
2
lim − tan x = ∞ and
x→ 3π
2
lim + tan x = −∞
x→ 3π
2
Intercepts:
Putting all of this together gives the tangent graph: