P.o.D. – Simplify using trigonometric
identities.
2
1.) If sin 𝜃 = , find cos 𝜃.
3
2.) If sec 𝜃 = 2, find tan 𝜃.
3.) If tan 𝜃 = 3, find cos 𝜃.
2 2
1.) ( ) + 𝑐𝑜𝑠 2 𝜃 = 1 →
3
2
4
5
9
9
𝑐𝑜𝑠 𝜃 = 1 − = ;
5 √5
cos 𝜃 = √ =
9
3
2.) 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃
→ 𝑡𝑎𝑛2 𝜃 = 4 − 1 = 3
→ tan 𝜃 = √3
3.) 1 + 9 = 𝑠𝑒𝑐 2 𝜃 → 10 = 𝑠𝑒𝑐 2 𝜃 →
√10 = sec 𝜃 → cos 𝜃 =
1
√10
=
√10
10
4.4 – Trigonometric Functions of Any
Angle.
Learning Target: be able evaluate trig
functions of any angle; be able to use
reference angles.
Consider an angle in standard position
with (x,y) a point on its terminal side:
(x,y)
r
𝜃
𝑦
sin 𝜃 =
𝑟
𝑦
tan 𝜃 =
𝑥
𝑟
sec 𝜃 =
𝑥
Where 𝑟 = √𝑥 2 + 𝑦 2
𝑥
cos 𝜃 =
𝑟
𝑥
cot 𝜃 =
𝑦
𝑟
csc 𝜃 =
𝑦
EX: Let (-2,3) be a point on the terminal
side of 𝜃. Find the sine, cosine, and
tangent of 𝜃.
Begin by finding r.
𝑟 = √(−2)2 + 32 = √4 + 9 = √13
Now apply the formulas from above.
𝑦
3
3√13
sin 𝜃 = =
=
𝑟 √13
13
𝑥
−2
−2√13
cos 𝜃 = =
=
𝑟 √13
13
𝑦
3
tan 𝜃 = =
𝑥 −2
Sign of Trigonometric Values By
Quadrant:
Quadrant II:
Sin (+)
Cos (-)
Tan (-)
Quadrant I:
Sin (+)
Cos (+)
Tan (+)
Quadrant III:
Quadrant IV:
Sin (-)
Sin (-)
Cos (-)
Cos (+)
Tan (+)
Tan (-)
All Students Try Calculus!
4
EX: Given sin 𝜃 = and tan 𝜃 < 0, find
5
cos 𝜃 and csc 𝜃.
Since sine is positive and tangent is
negative, we can determine that our
angle is in Quadrant II. This means that
cosine will be negative and cosecant will
be positive.
We can find cosine by using a
Pythagorean Identity.
2
4
𝑐𝑜𝑠 2 𝜃 + ( ) = 1 →
5
16
9
9
𝑐𝑜𝑠 𝜃 = 1 −
=
→ cos 𝜃 = ±√
25 25
25
2
3
=−
5
We can find cosecant by using a
reciprocal identity.
1
1
5
csc 𝜃 =
=
=
sin 𝜃 4⁄
4
5
Some special cases that we need to pay
particular attention to are situations
when a point lies on one of the axes. See
Figure 4.38 on page 313 for a graphical
representation.
EX: Evaluate the cosecant and cotangent
𝜋
3𝜋
2
2
functions at 0, , 𝜋, 𝑎𝑛𝑑
.
The point (1,0) lies on the angle 0
radians.
𝑟 1
csc 𝜃 = = = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑦 0
𝑥 1
cot 𝜃 = = = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑦 0
𝜋
The point (0,1) lies on the angle .
2
𝑟 1
csc 𝜃 = = = 1
𝑦 1
𝜋 𝑥 0
cot = = = 0
2 𝑦 1
The point (-1,0) lies on the angle 𝜋.
𝑟 1
csc 𝜋 = = = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑦 0
𝑥 −1
cot 𝜋 = =
= 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑦
0
The point (0,-1) lies on the angle
3𝜋
2
.
3𝜋 𝑟
1
csc
= =
= −1
2
𝑦 −1
3𝜋 𝑥
0
cot
= =
=0
2
𝑦 −1
Reference Angle:
- If an angle is in standard position, its
reference angle is the acute angle
formed between the terminal side
and the x-axis.
- Draw an example of a reference
angle in each quadrant on the
whiteboard.
EX: Find the reference angle for 𝜃 =
213°.
RA=213-180=33 degrees.
EX: Find the reference angle for 𝜃 =
1.7 𝑟𝑎𝑑𝑖𝑎𝑛𝑠.
𝑅𝐴 = 𝜋 − 1.7 = 1.44159 radians
EX: Find the reference angle for 𝜃 =
144°.
RA=180-144=36 degrees
*Memorize the “Trigonometric Values of
Common Angles” on the bottom of page
315.
𝜃 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 0°
𝜃 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 0
sin 𝜃
0
cos 𝜃
1
tan 𝜃
0
30°
𝜋
6
½
45° 60° 90°
𝜋
𝜋
𝜋
4
2
3
√2 √3 1
2
2
0
√3 √2 ½
2
2
√3 1
√3 Undef
3
5𝜋
EX: Evaluate sin
180° 270°
3𝜋
𝜋
2
0
-1
-1
0
0
undef
3
We must recognize that
5𝜋
3
is in
Quadrant IV. Therefore, we need to find
its reference angle.
𝑅𝐴 = 2𝜋 −
5𝜋
3
=
6𝜋
3
−
5𝜋
3
𝜋
= .
3
Now use our memorized chart to find
the sine of the reference angle.
𝜋 √3
sin =
3
2
However, we already indicated that this
angle is in Q4, so we know that our value
must be negative. Thus,
5𝜋
√3
sin
=−
3
2
EX: Evaluate cos(−60°)
We begin by finding either a reference
angle or a coterminal angle.
Coterminal: −60° + 360° = 300°
Reference: 360° − 300° = 60°
Now find the cosine of 60 degrees using
our memorized chart.
1
cos 60° =
2
300 degrees is in Q4, and cosine is
positive in Q4. Therefore, the answer
remains ½ .
EX: Evaluate tan
Recognize that
5𝜋
6
5𝜋
6
is in Quadrant II.
Its reference angle is 𝜋 −
5𝜋
6
𝜋
= .
6
𝜋
From our chart, we know that tan =
6
√3
.
3
In Q2, tangent is negative, so our
final answer is
√3
− .
3
*We can also solve trigonometric
functions using our identities.
EX: Let 𝜃 be an angle in Quadrant III such
5
that sin 𝜃 = − . Find sec 𝜃 and tan 𝜃
13
by using trigonometric identities.
We will start by finding cosine using a
Pythagorean Identity.
2
5
𝑐𝑜𝑠 𝜃 + (− ) = 1 →
13
25
144
2
𝑐𝑜𝑠 𝜃 = 1 −
=
169 169
2
144
12
→ cos 𝜃 = ±√
=−
169
13
Cosine must be negative since we are in
Q3.
We can now find secant by using a
reciprocal identity.
1
1
13
sec 𝜃 =
=
=−
cos 𝜃 −12⁄
12
13
We can find tangent by using a Quotient
Identity.
−5⁄
sin 𝜃
−5 13
13
tan 𝜃 =
=
=
∙
cos 𝜃 −12⁄
13 −12
13
5
=
12
EX: Use a calculator to evaluate each
trigonometric function.
a.) cot 375°
b.) sin(−4.1)
c.) sec
3𝜋
8
*Always remember to check your mode
(radians or degrees).
a.)
b.)
c.)
Upon completion of this lesson, you
should be able to:
1. Find the trig functions for any point.
2. Use reference angles to find trig
functions.
3. Apply identities to find trig
functions.
For more information, visit
http://academics.utep.edu/Portals/1788/CALCULUS%20
MATERIAL/4_4%20TRIF%20FNS%20OF%20ANGLES.pdf
HW Pg.318
106E
6-84 6ths, 88, 92, 98-
Quiz 4.1-4.4