TRIGONOMETRY - · PDF file · 2012-02-21Trigonometry P1 1 MES LAILA PTEK 2011 ......

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  • Trigonometry P1 1

    MES LAILA PTEK 2011

    TRIGONOMETRY

    Trigonometric Ratios

    Example 1

    Given the right-angled triangle ABC and that tan = 2, find sin and cos .

    ----------------------------------------------------------------------------------------------------------------------------------

    Trigonometric Ratios of Some Special Angles

    The trigonometric ratios of angles measuring 30o, 45o and 60o can be obtained using the right-angled

    triangles formed by taking half of a square and an equilateral triangle.

    Trig / 0o 30o 45o 60o 90o

    Cos

    Sin

    Tan

    Trigonometric Ratios of Complementary Angles

    In the triangle PQR in the diagram, P and R are complementary angles as P + R = 90o.

    Example 2

    Deduce the value of

    O

    Q

    P Sin =

    Cos =

    Tan =

    Cos (90o )=

    Sin (90o ) =

    Tan (90o- ) =

    R Q

    P

    (90o- )

  • Trigonometry P1 2

    MES LAILA PTEK 2011

    General Angles

    In general, the angle is measured from the positive x- axis.

    (a) is positive if it is measured in an anti-clockwise direction.

    (b) is negative if is measured in a clockwise direction.

    (c) For positive , the quadrant of is determined by the value of .

    The following four diagrams show some values of in the four quadrants, with a basic angle = 45o. Observe that all the angle are in the anticlockwise sense as indicated by the arrowheads and the values of are said to be positive.

    In the following four diagrams, = 45o. Observe that the angles are negative in value because they are

    in the clockwise sense as indicated by the arrowheads.

    y

    x =30o

    = - 30o

    anticlockwise

    clockwise

    2nd

    Quadrant 90

    o <

  • Trigonometry P1 3

    MES LAILA PTEK 2011

    Example 3 Given that 0o < < 360o and has a basic angle of 40o, find if it is in the

    (a) 3rd quadrant (b) 4th quadrant ------------------------------------------------------------------------------------------------------------------------------------------

    Basic Angle

    is an acute angle. It is made by the arm of the line segment and the x-axis. The sine, cosine and

    tangent of a basic angle should be taken as positive. Below are the position of basic angle in the four

    quadrant :

    Signs of Trigonometric Ratios in the Four Quadrants

    1st Quad 2nd Quad 3rd Quad 4th Quad

    Sin +ve +ve -ve -ve

    Cos

    Tan

    Example 4

    Without using a calculator, determine whether the following trigonometric ratios are positive or

    negative.

    (a) sin 230o (b) cos 140o (c) tan 215o

    (d) cos 350o (e) tan 340o (f) sin 160o

    (g) cos (-160o) (h) tan (-155o)

    All Sin

    Tan Cos

    2nd

    Quadrant

    = =

    3rd

    Quadrant Quadrant

    = =

    1st

    Quadrant

    4th

    Quadrant Quadrant

  • Trigonometry P1 4

    MES LAILA PTEK 2011

    Example 5

    Evaluate the following ( without using calculator) :

    (a) tan 300o (b) cos 330o (c) sin 150o (d) tan 315o

    (e) sin 225o (f) cos 210o (g) tan (-120o) (h) sin 405o

    Example 6

    Without using a calculator, evaluate the trigonometric ratios of 300o. Hence deduce the trigonometric

    ratios of 660o.

    Example 7

    Given that cos =

    and 180o< < 270o, evaluate tan and sin .

    Example 8

    Express the trigonometric ratios of 70o in terms of the ratios of its basic angle.

    The above results are particular cases of the following more general results :

    For any angle ,

    Cos (- ) =

    Sin (- ) =

    Tan (- ) =

  • Trigonometry P1 5

    MES LAILA PTEK 2011

    EXERCISE 5A TRIGONOMETRIC RATIOS AND GENERAL ANGLES

    1. For each of the following triangles, find the values of cos , sin and tan

    (a) (b) (c)

    2. For the given right-angled triangle below, complete the table

    of trigonometric ratios on the right :

    3. Given that sin =

    , find the value of sin cos (90o ).

    4. Given that tan A = 2, find the value of 2 tan A + tan (90o A).

    5. Without using a calculator, evaluate

    (a)

    (b) tan 45o + tan 30o tan 60o

    6. Without using a calculator, find the value of

    (a)

    (b) tan 75o tan 15o

    7. State the quadrant of the angle and find the value of the basic angle if

    (a) = 0 (a) = 90 (a) = 0 (a) = 00

    8. For each of the following diagrams, find the value of the angle .

    (a) (b) (c) (d)

    9. Find all the angles between 0o and 360o which make a basic angle of with the x-axis if

    (a) = 20o (b) = 70o (c) = 35o

    10. Find all the angles between 180o and 180o with basic angle = 10o.

    Cos Sin Tan

    (a)

    (b)

    (c)

    (d) 3

    y

    x o

    50o

    y

    x o 60o

    y

    x o 30o

    x o 45o

    y

  • Trigonometry P1 6

    MES LAILA PTEK 2011

  • Trigonometry P1 7

    MES LAILA PTEK 2011

    Example 2 : Effect of sin bx, cos bx, tan bx

    Sketch the graph of the following.

    a) y = sin 2x for 0o 180o

    b) y = cos x for 0o 360o.

    Example 3 : Effect of sin x + c , cos x + c, tan x + c

    Sketch the graph of the following 0 .

    a) y = sin x + 2

  • Trigonometry P1 8

    MES LAILA PTEK 2011

    b) y = cos x 3

    Example 4 : Effect of a sin bx + c , a cos bx + c, a tan bx + c

    Sketch the graph of the following 0o 360o.

    a) y = 2 sin 3x + 1

    b) y = 3 cos 2x - 2

    EXERCISE 5B : SKETCHING TRIGONOMETRIC GRAPHS

    1. Sketch on separate diagram, the following for 0o 360o and state the corresponding range of y. (a) y = sin 2x 1 (b) y = 4 sin 3x + 2 (c) y = 3 + 4 cos 2x (d) y =3 cos 2. Sketch on separate diagram, the following for the interval 0 . (a) y = 5 cos 2x (b) y = 1 + sin 2x (c) 4 sin 3x 2 (d) 3 cos 2x 1

  • Trigonometry P1 9

    MES LAILA PTEK 2011

    Solving Basic Trigonometric Equations

    Basic trigonometric equations of the form cos = , sin = and tan = , where k is a

    constant, are solved as follows :

    Example 1

    Find all the value of such that

    a) sin = - 0.5 where 0o < < 360o ,

    b) tan = -1 where 0o < < 360o ,

    c) cos = 0.46 where -360o < < 0o.

    Example 2

    Find all the angles between 0o and 360o which satisfy the equation

    (a) cos (x + 30o) = -0.3 (b) sin 2x = 0.866

    (c) tan (2x 50o) = -0.7 (d) cos2 x =

    (e) sin2 x + sin x cos x = 0 (f) 6 cos2 x cos x 1 = 0

    Example 3

    Solve 3(tan x + 1) = 2 for -180o x 180o.

    Example 4

    Solve the following equations for 0 , leaving your answers in terms of .

    (a) 4 cos2 x = 3 (b) 1 2 sin2x = 3 sin x + 2

    EXERCISE 5C : SOLVING BASIC TRIGONOMETRIC EQUATIONS

    Step 1 : Identify the quadrant by looking at the sign of k,

    Step 2 : Find the basic angle by taking the positive value of k only,

    Step 3 : Find all the values and the limit of .

  • Trigonometry P1 10

    MES LAILA PTEK 2011

    1. Find all the angles x where 0o < x < 360o such that

    (a) cos x = - 0.71 (b) tan x = 1.732 (c) sin x = 0.866 (d) tan x = -2

    (e) 10 cos x 3 = 0 (f) 4(tan x 1) = 3(5 2 tan x) (g) 2 sin (-x) = 0.3 (h) 2 cos2 x = 1

    (i) 3 sin x + 2 = tan 75o (j)

    = 3

    2. Find all the angles between 360o and 180o such that

    (a) sin x = - 1/2 (b) cos x = 3/2 (c) tan (-x) + 1 = 0 (d) 2 sin (90o 1) + 1 = 0

    3. Positive angles x and y are such that x + 2y = 300o and tan y = 2 cos 160o. Find their values. 4. Find the values of x, where 0 , which satisfy each of the following equations.

    (a) cos x =

    (b) tan x = 1

    (c) sin x = - 0.5 (d) 6 sin2x 1 = 0

    (e) 2 sin x cos x = cos x cos 40o (f) 10 cos 2x = 7

    5. Solve the following equations for all values of x from 180o to +180o.

    (a) cos (x 20o) =

    (b) cos x ( sin x 1) = 0

    (c) 3 sin2x = 2 sin x cos x (d) 2 cos2 5 cos x + 2 = 0

    ------------------------------------------------------------------------------------------------------------------------------------

    ANSWERS

    `1. (a) 135.2o , 224.8o (b) 60o , 240o (c) 60o, 120o (d) 116.6o, 296.6o

    (e) 72.5o, 287.5o (f) 62.2o, 242.2o (g) 188.6o, 351.4o (h) 45o, 135o, 225o,315o

    (i) 35.3o, 144.7o (j) 63.0o, 297.0o

    2. (a) - 1500, - 30o (b) - 330o, - 30o, 30o (c) -315o, -135o, 45o (d) -225o, -135o, 135o

    3. x = 64.0o, y = 118.0o.

    4. (a)

    (b)

    (c)

    (d) 0.421, 2.72, 3.56, 5.86

    (e)

    , 0.393, 2.75 (f) 0.398, 2.74, 3.54, 5.89

    5. (a) -115o, 155o (b) 90o (c) 0, 33.7o, -146.3o, 180o (d) 60o

  • Trigonometry P1 11

    MES LAILA PTEK 2011

    Simple Identities

    The identities shown are commonly used to :

    (a) Solve a trigonometric equation which could