Trigonometric Functions II – Graph of SINE & COSINE of Angles SS2 Further Mathematics Lesson Note

GRAPH OF SINE AND COSINE FOR ANGLES

Draw a circle on a Cartesian plane so that its radius, OP, is of length 1 unit. 

N.B: Such a circle is called a unit circle.

The angle Ѳ that OP makes with Ox changes according to the position of P on the circumference of the unit circle. Since P is the point (x,y) and /OP/ = 1 unit,

         Sin Ѳ = y/1     = y

          Cos Ѳ = x/1   = x

Hence the values of x and y give a measure of cos Ѳ and sin Ѳ respectively.

If the values of Ѳ are taken from the unit circle, they can used to draw the graph of sin Ѳ. This is done by plotting values of y against corresponding values of Ѳ

When drawn, you will see that the vertical dotted lines give the values of sin Ѳ corresponding to Ѳ = 30, 60,90,……., 360.

To draw the graph of cos Ѳ, use corresponding values of x and Ѳ. This gives another wave-shaped curve.

As Ѳ increases beyond 360, both curves begin to repeat themselves as in the figures below.

Notice the following:

1)All values of sin Ѳ and cos Ѳ lie between +1 and -1.

2)The sine and cosine curves have the same shapes but different starting points.

3)Each curve is symmetrical about its peak(high point) and trough(low point). This means that for any value of sin Ѳ there are usually two angles between 0 and 360; likewise cos Ѳ. The only exceptions to this are at the quarter turns, where sinѲ and cosѲ have the values given in the table below;

Examples

1. Referring to the graph on page 211 of New General Mathematics Book 1, 

(a)Find the value of sin 252, 

b)solve the equation   5 sin Ѳ = 4

Solution

a)On the Ѳ axis, each small square represents 6. From construction a) on the graph:

        Sin 252 = -0.95

b) If 5 sin Ѳ = 4

   then sin Ѳ = 4/5  = 0.8

  From construction (b) on the graph: when sin Ѳ = 0.8, Ѳ = 54 or 126

Graph of Tan Ѳ

Values can be taken from a unit circle to draw a tangent curve. Draw a tangent to the unit circle OX. 

Draw a radius and extend it to meet the tangent at T. The – coordinates of T give a measure of tan Ѳ, where Ѳ is the angle that the radius makes with OX.

Note that tan Ѳ is not defined when Ѳ =90° and 270°.

Ratio of Special Angles (45°, 30° and 60°)

1. Tan, Sin and Cos of 45°

If ABC is a right-angled triangle at B and /AB/ = /BC/ = 1 unit

/AC/2 = 12 + 12  = 2 (by Pythagoras’ theorem)

/AC/ =

Thus, tan45° = 1

          Sin45°= 

          Cos45° = 

1. Tan, Sin and Cos of 30° and 60°

If ABC is an equilateral triangle with sides of 2 units in length. Line AD is an altitude where /BD/ = /DC/ = 1 unit.

In ABD, /AB/2 = /AD/2 + /BD/2 (by Pythagoras’ theorem)

22 = /AD/2 + 12

/AD/2 = 3

/AD/ =  units

Since, <B = 60°

Thus, Tan 60°=

           Sin 60°= 

           Cos 60°= ½  = 0.50

Also, <BAD = 30°

          Tan 30° =

           Sin 30°= ½  = 0.50

           Cos 30°=

Example: Write the value of each of the following in surd form;

1. sin135°
2. tan330°
3. cos240°

Solution 

1. sin135π = sin(180 -135) = sin 45 =   = 
2. tan330° = -tan(360-330) = tan30 =   
3. cos240° = cos(240 -180) = cos60 = – ½