Unit 4: Trigonometry I (Day 1)

Angles in Standard Position

Consider a point P (x, y) moving around a circle drawn on a grid with center

(0,0) and radius r . We wish to measure the angle between the line joining P to

the origin and the positive x-axis. We say that this angle is in standard position.

Some conventions for angles measured in standard position:

1. The positive x axis is _____

2. Angles have their vertex at _______

3. Positive angles are measured ___________________ (ccw)

4. Negative angles are measured ________________(cw)

5. The arm on the positive x-axis is called the _______________

6. The other arm is called the _______________

Consider a circle of radius 1 (called a unit circle).

Degree Measure

The measure you are familiar with is degrees:

- One full rotation is 360

- 1

360 of a rotation has a measure of 1 degree

Draw:

i) 60 o ii) -40 o iii) -490 o

Radian Measure

Definition:

1 radian is the measure of an angle containing an arc of length r in a circle of

radius r.

A circle has 360 . The circumference of a circle is 2C r=

When 1r = → 2C =

Then one complete rotation about the unit circle is an arc length of 2 for

every 360 .

So…… 2 radians 360o = or 180o =

Draw:

iv) 3

v)

4

vi)

4

3−

vii) 6

5 viii)

6

7 ix)

3

5−

Conversion Factors:

Degrees → Radians: Multiply by o180

Radians → Degrees: Multiply by

o180

Examples:

Convert from degrees to radians:

i) 30o ii) 90o

iii) 45o iv) 110o

v) 360o v) 270o

* We often leave radians in terms of

** The units for radians are either: ‘radians’ or nothing.

If an angle measures 2 then we are working in radians.

Convert from radians to degrees.

i) 1 radian ii) 5.6 radians

iii) 6

5 iv)

4

3

Coterminal Angles:

Two or more angles in standard position with the same P position.

Find two angles coterminal with 40 o:

Find two angles coterminal with 4

:

Given any angle , any coterminal angle is represented by:

In Degrees:

In Radians:

Examples:

a) Draw the given angle

b) draw 2 coterminal angles and find their measurements

i) o150= b) 3

−

Finding the length of an arc in a circle of radius r:

1. Determine the fraction of the circle being worked with.

2. Multiply this by the circumference

Degrees: Arclength = rao

2360

=

Radians: Arclength ra

2

2=

Examples:

i) A circle has radius 8 cm. Calculate the length of the arc subtended by each

angle:

a) 2.3 radians b) 75o

ii) A circle has radius 5 cm. Find the angle at the center containing an arc length

of:

a) 6 cm (in degrees) b) 15 cm (in radians)

Homework:

Workbook p. 255 #1,4-11

WS 4-1 and 3.3

Unit 4: Trigonometry I (Day 2)

Sine and Cosine in Standard Position

Recall:

Given a right triangle, there are 3 primary trig ratios and 3 secondary

trig ratios:

Sine:

sin opposite

hypotenuse =

Cosecant:

1csc

sin

hypotenuse

opposite

= =

Cosine:

cos adjacent

hypotenuse =

Secant:

1sec

cos

hypotenuse

adjacent

= =

Tangent:

tanopposite

adjacent =

Cotangent:

1cot

tan

adjacent

opposite

= =

These definitions are for angles 0 90 . We need to expand our

definitions for 90 .

Consider P(x, y) a point on terminal side of , in a circle of radius r.

The values of ,x y and r determine the six trig ratios for angle :

sin = csc =

cos = sec =

tan = cot =

The quadrant in which is found will determine whether the trig ratio

is /+ − :

Ex:

i) In which quadrant is sin < 0 and tan > 0?

ii) In which quadrant is cos > 0 and tan <0?

Ex:

i) Determine cos if 41

4sin −= and tan 0

ii) Determine sin and cos if is an angle in standard position

whose terminal arm side is the graph 2 5 0, 0x y x+ =

iii) Given the point (1, 2)− on the terminal arm of an angle in

standard position, determine the value of all 6 trig ratios.

p. 262 # 1-3,5,6(a-d),7

WS 4-5

Unit 4: Trigonometry I (Day 3)

General and Special Angles

Learning

Intention(s): Determine the ratio of special angles

Solve a primary trig ratio for an unknown angle in both degrees and radians

Special cases: 0, 90, 180, 270, …

0 90 180 270

sin

cos

tan

There are 2 right triangles with ‘nice’ angle and side values.

45 o - 45 o - 90 o 30 o - 60 o - 90 o

0 o 30 o 45 o 60 o 90 o

sin

cos

tan

0 6

4

3

2

Reference Angles:

The acute angle formed by the terminal arm and the x-axis.

Examples: Find the reference angles for the following.

i) o220= ii) o410−=

iii) 3

2 iv)

6

5−

v) 4

7 vi)

12

7−

Determining the exact value of trig functions given angles in both degrees and radians:

1. Determine the reference angle

2. Determine the quadrant

Examples: Determine the following as exact values:

i) sin 240𝑜 ii) cos 120𝑜

iii) 4

5tan

− iv)

6

11cos

v) 13tan vi) 2

15sin

vii) 4

29tan

viii)

3

7cos

In reverse: Solve for if 3600

2

1cos = 2sin2 =

3

3tan −=

Solve for if 20

3sin2 = 2

2cos −=

Homework:

6.3 # 1-7odd, 11

Unit 4: Trigonometry (Day 4)

Graphing the Primary Trig Functions

We can graph in either degrees or radians. Radians are generally easier to count.

1. sin)( =f

0 6

2

65

67

23

611 2

sin

Properties of sin:

1. Domain 2. Range

3. Period 4. Amplitude

5. -intercepts: 6. y-intercept:

7. Maximum Value: _______ At what values of does the maximum occur? _________

8. Minimum Value: _______ At what values of does the minimum occur? __________

2. cos)( =f

0 3

2

32

34

23

35 2

cos

Properties of cos:

1. Domain 2. Range

3. Period 4. Amplitude

5. -intercepts: 6. y-intercept:

7. Maximum Value: _______ At what values of does the maximum occur? _________

8. Minimum Value: _______ At what values of does the minimum occur? __________

3. tan)( =f

0 4

2

43

45

23

47 2

tan

Properties of tan:

1. Domain:

2. Range:

3. Period:

4. -intercepts:

5. y-intercept:

6. Asymptotes:

Homework:

Practice sketching 1 period of the Sine, Cosine and Tangent functions.

Unit 4: Trigonometry (Day 5)

Graphing y = asin(x – p) + q and y = acos(x – p) + q

a : Expands/Compresses the graph vertically

p: Translates the graph left/right *also called the phase shift

q: Translates the graph up/down *also called the vertical displacement

Examples: Graph the following, then state the given properties

i) 1)sin(33+−= xy

Domain: Range:

Amplitude: Period:

ii) 2)cos(26++= xy

Domain: Range:

Amplitude: Period:

Phase Shift: Vertical Displacement:

iii) 1)sin(22−+−= xy

Domain: Range:

Amplitude: Period:

Example:

The following is a graph of a sine function. Write a possible equation.

The above could also be a cosine function. Write a possible equation.

Graphs of this shape are said to be sinusoidal. The equation of a

sinusoidal can be either sine or cosine.

Homework: WS (p. 245) #1, 3, 14, 15, 18, 19adgjmn, 25ade

Unit 4: Trigonometry (Day 6)

Graphing y = a sin b (x – p) + q

b: Expands/Compresses the graph horizontally by a factor of 1/b.

For a sin/cos graph, the b value will affect the period. Period b

T2

=

Examples: Graph the following:

i) xy21sin=

ii) xy 2cos=

iii) 3)(3sin26−−= xy

iv) 1)cos(342

1 ++−= xy

Domain: Range:

Amplitude: Period:

v) The following is a cosine function. Write a possible equation.

WS (p. 254) #1, 2, 3d, 7bc, 9bf, 11cd, 12bd, 14, 15, 17ad

(answer to 3d is 4 to the right)

Unit 4: Trigonometry (Day 7)

Graphing xyp2sin= and xy

p2cos=

Recall: The period of sine/cosine is T = b

2

Examples: Determine the period, T. Then sketch the graph

i) xy5

2sin =

ii) xy1002cos =

iii) 52sin34

)2(−=

−xy

Example:

The volume of air in one’s lungs is a sinusoidal function of time. A graph showing normal

breathing is shown below. Determine the equation for this function:

Homework:

WS (p. 265) #1-3, 5ad, 7ab, 14cd, 15ab, 20

2700

2200

Time (s) 0 5 10 15

Volume (mL)

Unit 4: Trigonometry (Day 8)

More Graphing y = a sin b (x – p) + q

Graph 1)tan(34−−= xy

Period: Asymptotes:

Domain: Range:

Unlike sine and cosine, xy tan= has a period of:

The period of bxy tan= is therefore:

Graph 1)(tan421 +−= xy

Period: Asymptotes:

Domain: Range:

Graph: )1(4

tan3 −= xy

Domain: Range:

Graph 1)cos(342

1 ++−= xy

Domain: Range:

Amplitude: Period:

The following is a cosine function. Write a possible equation.

i) Write the equation for a sine function with amplitude 6, period 8,

phase shift -5 and vertical displacement 10

ii) Write the equation of the cosine function with maximum 18,

minimum 4, period 6 and phase shift 5

iii) Given 2)3(2cos5 −+−= xy

Determine the following:

Domain Range Period

Maximum value Minimum value

and when it first occurs and when it first occurs

iv) Write the equation of the sinusoidal function with a maximum of 8 at

when x is 2 and followed by a minimum of 1 when x is 9.

Homework: