BAM FORM 6 – TRIGONOMETRY

CIRCLE

Length of an Arc

Consider the following

    

2 (length of arc AB) = θ x 2r

                        Length of arc AB = rθ

     Area of a circular sector

            Pg. 141 drawing

             =

            Where: area of circle = r² and 360^o = 2

             =

            Area of sector AOB x 2 = θ x r²

          

             Area of sector AOB = r²θ

Where θ is in radians and r is the radius of the circle

Example 1.15

Find the arc length of a circle of radius r which is subtended by a central angle θ for each of the following

a) r = 3cm and θ = 45^o

b) r = 6cm and θ = 180^o

Use  = 3.14

Solution

a) 1 = rθ

            1 = 3 x cm or =  = 2.354 cm

b) arc length, 1 = r θ

                          1 = 6 x

                          1 = 6 x 3.14

                           = 18.84cm

Example 1.16

Find the area of the sector of a circle with radius 6cm and subtended by an angle of 60^o use  = 3.14

Solution

Area =  r²θ

         = ½ x 6 x 6

         = 6  cm²

Since = 3.14

Therefore, area = 3.14 x 6

                          = 18.84cm²

Trigonometric Equation

Example 1.17

Solve sin θ = ½ for 0^o ≤ θ ≤ 360^o

Solution

Sin θ = ½
θ = Sin^-1 ( ½)
θ = 30^o, 150^o

Example 1.18

Solve 2sin θ –  = 0 for 0 ≤ θ ≤ 360

Solution

2 sin θ =

Sin θ =

θ = 60^o, 120^o

Example 1.19

2cos² θ + 3cos θ – 2 = 0

Solution

This equation is quadratic in cos θ

Factorize the equation

2cos² θ + 3cos θ – 2 = 0

            Let cos θ = t

            2t² + 3t – 2 = 0

            2t² + 4t – t – 2 = 0

            2t (t + 2) – (t + 2) = 0

            (t + 2) (2t – 1) = 0

            t = -2 or 2t = 1

                        t = -2 or t = ½

But, t  = Cos θ

Cos θ = -2

θ = undefined or no solution since the minimum value of cos θ = -1

Cos θ = ½ = θ = 60^o, 300^o

θ = (60^o, 300^o)

Example

Solve Cos 2 θ Cos θ + Sin 2θ = 1, where 0≤ θ ≤ 360^o

Solution

Use the compound angle formula Cos (A – B) = Cos A Cos B + Sin A Sin B

Thus, Cos 2 θ Cos θ + Sin θ = Cos (2θ – θ) = Cos θ

            Cos θ = 1
θ = (0^o, 360^o)

Trigonometric functions

Let x be angle in degree or radians and f (x) be the image of x. Then the functions defined as f (x) = sin x, f (x) = cos x and f (x) =tan are called trigonometric functions. These functions can be graphed just like any other functions. Values of angles should be on the horizontal axis and images values should be on the vertical axis.

Sine functions f (x) = sin x

To graph sine functions, draw a table of values in which x values will be angles and y = f (x) will be images of these angles. The angles may be in degrees or radians. Table 8.1 has values that suit a function y = sin x, (The y – values are correct to one decimal place)

Cosine function y = cos x

To graph cosine function follow the procedure followed in drawing sine functions. Table 8.2 contains values for y = cos x. In table 8.2 values are given to one decimal place.

           

Note

1.  The domain of y = sin x and y = cos x is set of real numbers and range is     (-1 ≤ y ≤ 1).

2.  These functions are periodic, they repeat after every one complete   revolution (360^o)

            Graph of y = tan x

            To graph y = tan x draw a table values and plot the point required then join the points

with given correct to one decimal places.

           

Note

1.   The range of y = tan x is a set of real numbers

2.   The domain is (x: x ≠ 90^o, x ≠ 270, x ≠ 450…)

3.    The tangent function is periodic too. It repeats itself after every 180^o

4.   At x = 90^o, x = 450^o etc, the functions not defined. These values are its asymptotes. It tends to approach these values but never touches them

Exercise

1.  State the range and period for each of the following

            a) y = cos x

            b) y = ½ cos 2x

            c) y = ½ cos x

            d) y = 3 sin x

            e) y = 3 – sin x

            j) y = 1 + sin