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  • Chapter 4- Trigonometry

    Lesson Package

    MCR3U

    1

  • Table of Contents Lesson 1: Special Angles ...pg. 3-6 Lesson 2: Special Angles and Obtuse Angles.pg. 7-9 Lesson 3: Related and Co-terminal Angles.pg. 10-13 Lesson 4: Reciprocal Trig Ratios..pg. 14-15 Lesson 5: Problems in 2 Dimensions.....pg. 16-18 Lesson 6: Problems in 3 Dimensions..pg. 19-21 Lesson 7: Ambiguous Case of Sine...pg. 22-24 Lesson 8: Trig Identities 1....pg. 25-27 Lesson 9: Trig Identities 2....pg. 28-29

    2

  • Special Angles

    Trigonometry Review

    - There are three primary trigonometric ratios for right angled triangles.

    Sine, Cosine, and Tangent

    If we know a right angle triangle has an angle of ' ', all other right angle triangles with an angle of ' ' are similar and therefore have equivalent ratios of corresponding sides.

    sin

    cos

    tan =

    =

    = opposite

    opposite

    hypotenuse

    hypotenuse

    adjacent

    adjacent

    Acronym:

    S TCO OH HA ASOH CAH TOA

    1x

    a) Finding a missing side

    DO IT NOW!

    3

  • b) Finding a missing angle

    2 The idea of Special Angles:

    We look at two types of right triangles

    i) isosceles: 45 - 45 - 90 ii) equilateral (split in half): 30 - 60 - 90

    We get EXACT values for the three primary trig ratios using these special triangles -

    Special Angles

    Special Angles1) A triangle

    45O

    3 primary trig ratios

    Special Angles2) A triangle

    2

    Note: this is an equilateral triangle

    3 primary trig ratios

    4

  • Example 1: Use special triangles to find the exact values of all sides and angles:

    4cm

    Example 2: Use special triangles to find the exact values of all sides and angles:

    Example 3: Determine the exact value of

    Rationalizing the Denominator(gets rid of the square root in the denominator)

    a)

    b)

    5

  • Example 4: Find the exact value of

    6

  • Special Angles and Obtuse AnglesHomework Questions?

    Review of Special Angles

    Isosceles 45-45-90 Half Equilateral 30-60-90

    45O

    45O

    1

    12 2

    1

    3

    Terminal Arm:

    The arm of an angle that meets the initial arm at the origin and rotates around the origin counterclockwise to form a positive angle (or clockwise to form a negative angle). It determines what is.

    Initial Arm:

    First arm of an angle drawn on a Cartesian plane that meets the other (terminal) arm of the angle at the origin. Is always at 0 degrees (does not move).

    7

  • Any angle over 90 has a reference angle. The reference angle is between 0 and 90 and helps us determine the exact trig ratios when we are given an obtuse angle (angle over 90 degrees). The reference angle is the angle between the terminal arm and the closest x-axis (0/360 or 180).

    120

    Using Reference Angles to Find Trig Ratios for Obtuse Angles

    60

    When finding the trig ratios of positive angles we are rotating counter clockwise from 0 degrees toward 360. The CAST rule helps us determine which trig ratios are positive in each quadrant

    C

    AS

    T

    all

    cosine

    sine

    tan

    CAST Rule

    Quadrant 1Quadrant 2

    Quadrant 3 Quadrant 4

    Example 1: Find the reference angle

    a) b)

    c) d)

    Example 2: Find the exact value of

    a) b)

    8

  • c) d)So far, we have talked about rotating counter clock wise. We can also rotate clock wise from the x-axis but the angles are negative.

    Negative Angles

    -45

    315

    -45 and 315 are the same angle.

    Example 3: Find the exact values of

    a) b)

    c)

    9

  • 4.2 Co terminal and Related Angles

    In this section, you will learn how to identify different angles that have the same trigonometric ratio, as well as learn how they are related.

    To do this you will have to visualize the terminal arm rotating around a circle centred at the origin of a grid with a radius of r. This is done so that we can extend our understanding of trig functions for a broader class of angles and see how different angles are related.

    http://www.mathsisfun.com/geometry/unit-circle.html

    x

    yr

    The circle being used has radius r. The radius and the coordinates of a point on the circle (x, y) are related to the primary trig ratios. Study the circle and write expressions for sin , cos , and tan in terms of x, y, and r.

    (x, y)

    DO IT NOW!

    sin =

    cos =

    tan =

    When finding the trig ratios of positive angles we are rotating counter clockwise from 0 degrees toward 360. The CAST rule helps us determine which trig ratios are positive in each quadrant

    C

    AS

    T

    all

    cosine

    sine

    tan

    CAST Rule

    Quadrant 1Quadrant 2

    Quadrant 3 Quadrant 4

    10

  • Ex. Given that and that lies in the first quadrant, determine exact values for and

    Example 1 Example 2

    a) Find both angles of for question 1 between 0 and 360 that have the same sine value

    b) Find another angle that will have the same cosine ratio as #1

    Example 3: Find both angles between 0 and 360 where sin = -0.7

    Example 4: Find both angles between 0 and 360 where tan=2.1

    11

  • Example 5: The point (-7, 19) lies on the terminal arm. Find the angle to the terminal arm. Find the reference angle.

    Co-terminal Angles: Angles in standard position that have the same terminal arm.

    Starting at 30 and rotating 360 counterclockwise will bring you back to the same terminal arm.

    30 + 360 = 390

    Therefore 30 and 390 are co-terminal angles.

    Note: you can also move clockwise and get negative angles as co-terminal angles.

    Example 6: Find three other co-terminal angles of 60

    Homework: 4.2 worksheet

    12

  • 13

  • 4.3 Reciprocal Trig Ratios

    DO IT NOW!Using a unit circle (circle with a radius of 1), determine the three primary trig ratios for each of the following angles.

    (0,1)

    (-1,0)

    (0,-1)

    (1,0)

    Remember:

    Reciprocal Trig Ratios

    Reciprocals: two expressions that have a product of 1.

    Review of Special Angles

    Isosceles 45-45-90 Half Equilateral 30-60-90

    45O

    45O

    1

    12 2

    1

    3

    14

  • Example 1: Use your knowledge of special angles to complete the following chart.

    Example 2: Determine the exact value of all six trig ratios for angle A.

    A

    B

    C 5m

    12m 13m

    Example 3: Each angle is in the first quadrant. Determine the measure of each angle.

    a) b)

    c)

    Example 4: Determine two angles between 0 and 360 that have a cosecant of -2

    15

  • 4.4 Problems in Two Dimensions

    Review

    Right triangle problems SOH CAH TOA

    Oblique triangle problemsSine Law

    Cosine Law

    Sine law

    Used when:

    i) two sides and an opposite angle are knownii) two angles and one side are known

    Cosine Law

    Used When:

    - two sides and a containedangle are known

    Used When:

    - all three sides are known

    16

  • Example 1: Jonathan needs a new rope for his flagpole but is unsure of the length required. He measures a distance of 10m away from the base of the pole. From this point, the angle of elevation to the top of the pole is 42. What is the height of the pole, to the nearest tenth of a metre?

    Example 2: Pam, Steven and Rachel are standing on a soccer field. Steven and Rachel are 23m apart. From Steven's point of view, the other two are separated by 72. From Pam's point of view, the others are separated by an angle of 55.

    a) Sketch a diagram

    b) Determine the distance from Pam to Rachel. Example 3: A drive belt wraps around three pulleys as shown. Find the perimeter of the drive belt to the nearest tenth of a cm.

    17

  • F

    G

    H5.1 cm

    4.8 cm

    6.2 cm

    Example 4: Find the measure of angle G

    Complete Worksheet

    18

  • 4.5 Problems in 3-D

    Review

    Right triangle problems SOH CAH TOA

    Oblique triangle problemsSine Law

    Cosine Law

    Sine law

    Used when:

    i) two sides and an opposite angle are knownii) two angles and one side are known

    Cosine Law

    Used When:

    - two sides and a containedangle are known

    Used When:

    - all three sides are known

    19

  • Example 1: A radio antenna lies due north of Sam's house. Sam walks to Elena's house, a distance of 1200m, 50 east of north. From Elena's house, the antenna appears due west, with an angle of elevation of 12. Determine the height of the antenna, to the nearest metre.

    Step 1: find the distance from Elena's house to the base of the flag pole.

    Step 2: find the height of the antenna.

    Example 2: Justine is flying her hot-air balloon. She reports that her position is over a golf course located halfway between Emerytown and Fosterville, at an altitude of 1500m. Fosterville is 16.0km east of Danburg, and Emerytown is 16.5km from Danburg, in a direction 42 south of east. What is the angle of elevation seen from Danburg, to the nearest degree.

    Step 1: find the measure of EF

    Step 2: find the measure of angle E

    20

  • Step 3: find the measure of DG

    Step 4: find the angle of elevation seen from Danburg.

    Note: EG is half of EF

    Example 3: A surveyor is on one side of a river. On the other side is a cliff of unknown height that she wants to measure. The surveyor lays out a baseline AB of length 150,. From point A, she selects pint C at the