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