4-3: Trigonometric Functions of Any Angle
What you’ll learn about ■ Trigonometric Functions of Any Angle ■ Trigonometric Functions of Real Numbers ■ Periodic Functions ■ The 16-point unit circle
. . . and why Extending trigonometric functions beyond triangle ratios.
Definitions of Trigonometric Functions of Any Angle
Let Ѳ be an angle in standard position with (x, y) a point on the terminal side of Ѳ and
Definitions of Trigonometric Functions of Any Angle
Let Ѳ be an angle in standard position with (x, y) a point on the terminal side of Ѳ and (x, y)
y
x
ropposite
adjacent
hypo
tenu
se
Definitions of Trigonometric Functions of Any Angle
Let Ѳ be an angle in standard position with (x, y) a point on the terminal side of Ѳ and (x, y)
y
x
r
adjacent
opposite
hypotenuse
𝑠𝑖𝑛𝜃=¿ 𝑦𝑟
𝑜𝑝𝑝h𝑦𝑝
=¿
𝑐𝑜𝑠𝜃=¿𝑎𝑑𝑗h𝑦𝑝
=¿𝑥𝑟
𝑡𝑎𝑛𝜃=¿
𝑐𝑠𝑐 𝜃=¿
𝑠𝑒𝑐𝜃=¿
𝑐𝑜𝑡 𝜃=¿
𝑜𝑝𝑝𝑎𝑑𝑗
=¿
h𝑦𝑝𝑜𝑝𝑝
=¿
𝑎𝑑𝑗𝑜𝑝𝑝
=¿
h𝑦𝑝𝑎𝑑𝑗
=¿
𝑦𝑥𝑟𝑦𝑟𝑥
𝑥𝑦
𝑥≠0
y
𝑥≠0
𝑦 ≠0
Let (-3, 4) be a point on the terminal side of Ѳ. Find the sin, cos and tan.(-3, 4)
y
x
r
adjacent
opposite
hypotenuse
x = -3 y = 4
𝑟=√𝑥2+𝑦2
𝑟=√(−3)2+(4 )2
𝑟=√9+16𝑟=±√25𝑟=5
r = 5
𝑠𝑖𝑛𝜃=𝑦𝑟¿
45
𝑐𝑜𝑠𝜃=𝑥𝑟¿−35
𝑡𝑎𝑛𝜃=𝑦𝑥¿
4−3
Positive and Negative QuadrantsQuadrant I
Quadrant III Quadrant IV
Quadrant IIx + y +
sinѲ +
cosѲ +tanѲ +
secѲ +cscѲ +
cotѲ +
x -
x - x +
y +
y - y -
sinѲ +
sinѲ - sinѲ -
cosѲ -
cosѲ - cosѲ +
tanѲ -
tanѲ + tanѲ -
cscѲ +
cscѲ - cscѲ -
secѲ -
secѲ - secѲ +
cotѲ -
cotѲ + cotѲ -
Given and 0<𝜃<
𝜋2
𝑥>0𝑦>0
𝜋2
<𝜃<𝜋
𝑥<0𝑦>0
𝜋<𝜃<3𝜋2
𝑥<0𝑦<0
3𝜋2
<𝜃<2𝜋
𝑥>0𝑦<0
Given and 0<𝜃<
𝜋2
𝑥>0𝑦>0
𝜋2
<𝜃<𝜋
𝑥<0𝑦>0
𝜋<𝜃<3𝜋2
𝑥<0𝑦<0
3𝜋2
<𝜃<2𝜋
𝑥>0𝑦<0
How do you get a negative? One is positive and one is negative
Given and 𝜋2
<𝜃<𝜋
𝑥<0𝑦>0
𝜋<𝜃<3𝜋2
𝑥<0𝑦<0
3𝜋2
<𝜃<2𝜋
𝑥>0𝑦<0
How do you get a negative? One is positive and one is negative
Given and 𝜋2
<𝜃<𝜋
𝑥<0𝑦>0
3𝜋2
<𝜃<2𝜋
𝑥>0𝑦<0
How do you get a negative? One is positive and one is negative
Given and 𝜋2
<𝜃<𝜋
𝑥<0𝑦>0
3𝜋2
<𝜃<2𝜋
𝑥>0𝑦<0
How do you get a negative? One is positive and one is negative
cosѲ +cosѲ -
cosѲ -cosѲ +
Given and
3𝜋2
<𝜃<2𝜋
y 𝑥>0
How do you get a negative? One is positive and one is negative
cosѲ +
𝑥=4𝑦=−5
𝑟=√𝑥2+𝑦2
𝑟=√(4)2+(−5)2
𝑟=√25+16𝑟=±√41𝑟=√41
𝑟=√41
𝑠𝑖𝑛𝜃=𝑦𝑟¿−5
√41
𝑐𝑜𝑠𝜃=𝑥𝑟¿
4
√41
𝑡𝑎𝑛𝜃=𝑦𝑥 ¿
−54
¿ −5√ 4141
¿ 4 √4141
Given and
3𝜋2
<𝜃<2𝜋
𝑥>0𝑦<0
How do you get a negative? One is positive and one is negative
cosѲ +
𝑥=4𝑦=−5𝑟=√41
𝑐𝑠𝑐 𝜃=𝑟𝑦¿ √41−5
𝑠𝑒𝑐𝜃=𝑟𝑥¿ √41
4
𝑐𝑜𝑡 𝜃=𝑥𝑦¿
4−5
𝑠𝑖𝑛𝜃=𝑦𝑟¿−5
√41
𝑐𝑜𝑠𝜃=𝑥𝑟¿
4
√41
𝑡𝑎𝑛𝜃=𝑦𝑥 ¿
−54
¿ −5√ 4141
¿ 4 √4141
Find sin and cos of 0, , 2
0
(1, 0)
𝜋
𝜋2
3𝜋2
(-1, 0)
(0, 1)
(0, -1)
𝑠𝑖𝑛0=𝑦𝑟
=¿01=¿0 𝑠𝑖𝑛
𝜋2
=𝑦𝑟
=¿11=¿1 𝑠𝑖𝑛 π=
𝑦𝑟
=¿01=¿0
𝑠𝑖𝑛3𝜋2
=𝑦𝑟
=¿−11
=¿-1
𝑐𝑜𝑠0=𝑥𝑟
=¿11=¿1
𝑐𝑜𝑠𝜋2
=𝑥𝑟
=¿01=¿0
𝑐𝑜𝑠 π=𝑥𝑟
=¿−11
=¿ -1
𝑐𝑜𝑠3𝜋2
=𝑥𝑟
=¿01=¿0
Ranges of Trigonometric Functions
• We know that • If the measure of increases toward
90o, then y increases• The value of y approaches r, and
they are equal when • So, y cannot be greater than r. • Using the convenient point (0,1) y
can never be greater than 1.
90o
siny
r
x
yr
90o
Ranges Continued
• Using a similar approach, we get:
1 sin 1
1cos 1
sec 1 sec 1
csc 1 csc 1
tan cot
or
or
and can be any real number
Determining if a Value is Within the Range
Evaluate (calculator)
(not possible)
(not possible)
cos 2
90o cot 0
3sin
2
Reference Angles
Reference Angle: the smallest positive acute angle determined by the x-axis and the terminal side of θ
ref angle ref angle
ref angle ref angle
Think of the reference angle as a “distance”—how close you are to the closest x-axis.
Definition of a Reference Angle
Let Ѳ be an angle in standard position. Its reference angle is the acute angle α formed by the terminal side of Ѳ and the horizontal axis.
Ѳα
α=180 - ⁰ Ѳα=π - Ѳ
Ѳ
αα=Ѳ - 180 ⁰α=Ѳ - π
Ѳ
αα=360 - ⁰ Ѳα=2π - Ѳ
Find the reference angle for Ѳ=300⁰
Ѳ
What quadrant is the terminal side in?
αα=360 - ⁰300⁰α=60 ⁰
α=360 - ⁰ Ѳ
Find the reference angle for Ѳ=2.3Ѳ
What quadrant is the terminal side in?
α α=3.14 – 2.3α≈ 0.8416
α= π - Ѳ
Find the reference angle for Ѳ=-135⁰
Ѳ
What quadrant is the terminal side in?
α Reference Angle : α= Ѳ - 180⁰
α=45 ⁰α=225 - 180⁰ ⁰
Find the positive coterminal angle to -135⁰
Coterminal angle =-135 + ⁰360 ⁰Coterminal angle = 225 ⁰
Common Trigonometric Functions
Ѳ(degrees) 0⁰ 30⁰ 45⁰ 60⁰ 90⁰ 180⁰ 270⁰Ѳ(radians)
sin Ѳ
cos Ѳ
tan Ѳ
0 𝜋6
𝜋4𝜋3
𝜋2
3𝜋2
π
0
1
0
12
√32
√33
√22
√22
1
√32
12
√3
1
0
und
0
-1
0
-1
0
und
Positive Trig Function Values
r
r
r
r
x-xy y
-y -y
ALLSTUDENTS
TAKECALCULUS
All functions are positive
Sine and its reciprocal are positive
Tangent and its reciprocal are positive
Cosine and its reciprocal are positive
Positive and Negative QuadrantsQuadrant I
Quadrant III Quadrant IV
Quadrant IIx + y +
sinѲ +
cosѲ +tanѲ +
secѲ +cscѲ +
cotѲ +
x -
x - x +
y +
y - y -
sinѲ +
sinѲ - sinѲ -
cosѲ -
cosѲ - cosѲ +
tanѲ -
tanѲ + tanѲ -
cscѲ +
cscѲ - cscѲ -
secѲ -
secѲ - secѲ +
cotѲ -
cotѲ + cotѲ -
Find the
Ѳ
What quadrant is the terminal side in?
α
α= Ѳ - πα= α=α=
Is cos positive or negative in quadrant III?
Positive Trig Function Values
r
r
r
r
x-xy y
-y -y
ALLSTUDENTS
TAKECALCULUS
All functions are positive
Sine and its reciprocal are positive
Tangent and its reciprocal are positive
Cosine and its reciprocal are positive
Positive and Negative QuadrantsQuadrant I
Quadrant III Quadrant IV
Quadrant IIx + y +
sinѲ +
cosѲ +tanѲ +
secѲ +cscѲ +
cotѲ +
x -
x - x +
y +
y - y -
sinѲ +
sinѲ - sinѲ -
cosѲ -
cosѲ - cosѲ +
tanѲ -
tanѲ + tanѲ -
cscѲ +
cscѲ - cscѲ -
secѲ -
secѲ - secѲ +
cotѲ -
cotѲ + cotѲ -
Find the
Ѳ
What quadrant is the terminal side in?
αα= Is cos positive or negative in quadrant III?
𝑐𝑜𝑠4𝜋3
=− cos𝜋3
− cos𝜋3
=−( 12 )
¿−12
Find the
Ѳ
What quadrant is the terminal side in?
α
α= 180 - ⁰150⁰α=30⁰
Is tan positive or negative in quadrant II?
Find the coterminal angle for -210⁰coterminal= -210 + ⁰360⁰coterminal= 150⁰
Positive and Negative QuadrantsQuadrant I
Quadrant III Quadrant IV
Quadrant IIx + y +
sinѲ +
cosѲ +tanѲ +
secѲ +cscѲ +
cotѲ +
x -
x - x +
y +
y - y -
sinѲ +
sinѲ - sinѲ -
cosѲ -
cosѲ - cosѲ +
tanѲ -
tanѲ + tanѲ -
cscѲ +
cscѲ - cscѲ -
secѲ -
secѲ - secѲ +
cotѲ -
cotѲ + cotѲ -
Find the tan
What quadrant is the terminal side in?α=30Is tan positive or negative in quadrant II?
𝑡𝑎𝑛 (−210 ° )=− tan 30 °
− tan 30 °=−( √33 )
¿− √33
Ѳα
Find the csc
Ѳ
What quadrant is the terminal side in?
α
α= - α= Is csc positive or negative in quadrant II?
Find the coterminal angle for coterminal= coterminal= coterminal=
= -
Positive and Negative QuadrantsQuadrant I
Quadrant III Quadrant IV
Quadrant IIx + y +
sinѲ +
cosѲ +tanѲ +
secѲ +cscѲ +
cotѲ +
x -
x - x +
y +
y - y -
sinѲ +
sinѲ - sinѲ -
cosѲ -
cosѲ - cosѲ +
tanѲ -
tanѲ + tanѲ -
cscѲ +
cscѲ - cscѲ -
secѲ -
secѲ - secѲ +
cotѲ -
cotѲ + cotѲ -
Find the csc
What quadrant is the terminal side in?α=Is csc positive or negative in quadrant II?
𝑐𝑠𝑐11𝜋
4=csc
3𝜋4
csc3𝜋4
=+csc𝜋4
¿√2
Ѳα
Finding Exact Measures of Angles
• Find all values of
• Sine is negative in QIII and QIV• Using the 30-60-90 values we found earlier, we know
3, 0 360 , sin
2o owhere when
3 sin 60
2o
Finding Exact Measures of Angles – Cont.
•
• Our reference angle is 60o. We must be 60o off of the closest x-axis in QIII and QIV.
3 sin 60
2o
240 300o oand
Note: there is other way to remember special angle, radian and point of unit circle