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

• 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 .

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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

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

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

Signs of Trigonometric Ratios in the Four Quadrants

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

= =

3rd

= =

1st

4th

• 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

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`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