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Ch 6.3: Trigonometric functions of any angle
In this section, we will
1. extend the domains of the trig functions (as well as the need
for it) using the unit circle
2. look at some properties of the unit circle
3. evaluate trig. values using reference angles
4. investigate the relationships between trig. functions
Why extend the right triangle definition?
1) 0 < θ since it’s an interior angle of a right triangle.
I
No sin(0◦ ) or cos(−50◦ ) can be defined through the right
triangle definition.
2) θ < 90 since we have a right triangle and the sum of the
interior angles in a triangle is 180.
I
I
In fact, if θ = 90, then we have two right angles. ⇒ No
triangle at all!!
If θ > 90, Same problem!!
Conclusion: With right triangle definition, we can define sine,
cosine, and tangent, but ONLY for θ ∈ (0, 90).
⇒ We need to extend our definition (and will use the unit circle to
do so..)
The Unit Circle definition - for sine, cosine, and tangent
Definition
Let (x, y ) be a point on the unit circle and θ is the angle between
the positive x-axis (initial side) and the ray from 0 which goes
through (x, y ) (terminal side). Then,
x = cos θ ,
y = sin θ ,
y
= tan θ.
x
Examples
Use the unit circle definition to evaluate sin θ and cos θ for the
following angles: θ = 0◦ , 90◦ , 180◦ , 270◦ , 360◦
Let’s check!
Let θ ∈ (0, 90) and (x, y ) be corresponding point on the unit circle.
New definition says cos θ = x and sin θ = y .
Q: If θ ∈ (0, 90), can we say the same using the old definition?
(Do we even have a right triangle?)
Properties of the unit circle (UC)
Since any point P(x, y ) on the unit circle (UC) has the following
relation:
x2 + y2 = 1
if we’re given one of x, y in (x, y ) on the UC, then we know the
other coordinate.
Example) If y = 12 , and (x, y ) in 2nd Q, what is x?
Properties of the unit circle (UC) - Continued
If point P(x, y ) on UC, then due to UC being symmetric to both x
and y axis (and also to the origin):
(x, −y ), (−x, y ), and (−x, −y )
are all on the UC as well.
−4
Example) If x = −3
5 and y = 5 , a) verify that (x, y ) is on the UC,
and b) find three other points on the UC?
Reference Angle
Definition
For a non-quadrantal angle θ in standard position, the acute angle
θr formed by the terminal side and the nearest x-axis is called the
reference angle.
Examples - Reference angles
Find the reference angle for a) θ = 315◦ , b) θ = 1900◦ , c)
θ = −120◦ .
More Examples
Find the reference angle for each rotation given.
a. θ = 5π
b. θ = 4π
c. θ = − 7π
6
3
4 .
Signs of the Trig functions
Knowing that cos θ = x and sin θ = y , (and tan θ = yx )
Ex1) Let sin α > 0 and tan α < 0. Which quadrant is α in?
Ex2) cos β < 0 and cot β > 0.
Examples - Evaluating trig. values using reference angles
Evaluate the following:
1. cos 330◦
2. sin( 7π
6 )
What if (x, y ) is NOT on the unit circle?
I
sin θ
I
cos θ
I
tan θ
I
csc θ
I
sec θ
I
cot θ
Definition
Definition
Given (x, y ) with r =
p
x 2 + y 2 and the corresponding θ,
sin θ =
csc θ =
y
,
r
cos θ =
x
,
r
tan θ =
y
x
1
r
1
r
1
x
= , sec θ =
= , cot θ =
=
sin θ
y
cos θ
x
tan θ
y
(Note: denominators must not be 0.)
Q: For which (quadrantal) angles are these functions not defined?
Examples
Find the values of the 6 trig functions, given the following.
1. (7, 24)
2. (−3, −1)
More example
Given sin θ = 5/13 and cos θ < 0, find the values of other ratios.
Fundamental Trigonometric Identities
Reciprocal Identities:
csc θ =
1
,
sin θ
sec θ =
1
,
cos θ
cot θ =
1
tan θ
Ratio Identities:
tan θ =
sin θ
sec θ
=
,
cos θ
csc θ
cot θ =
cos θ
sin θ
Cofunction Identities : From the unit circle definition
Theorem
cos θ = sin( π2 − θ), csc θ = sec( π2 − θ), cot θ = tan( π2 − θ)
Examples
Write the following in terms of its cofunction.
1. sin −60◦
2. cos π3
3. cot 120◦
4. sec π2
Homework for Ch 6.3: pg. 497
4, 18, 19, 44, 51, 65, 70, 71, 91, 92, 94