Mathematics Notes

Application of Surds to Trigonometrical Ratios Sine and Cosine graphs


Sine θ Cosine

The figure above shows the development of (a) sine graph   (b) cosine graph from a unit circle

Each circle has a radius of 1 unit. The angle θ that the radius OP makes with Ox changes as P moves on the circumference of the circles. Since P is the general point (x, y) and OP = 1 unit, then sin θ = y, Cos θ = x. Hence the values of x and y gives cos θ and sin θ respectively. These values are used to draw the corresponding sine and cosine curves. The following points should be noted on the graphs of sin θ and cos θ:

  1. All values of sin θ and cos θ lie between + 1 and – 1.
  2. The sine and cosine curves have the same wave shape but they start from different points. Sine θ starts from 0 while cosine θ starts from 1.
  3. Each curve is symmetrical about its crest(high point) and trough(low point). Hence, for the values of Sin θ and Cos θ there are usually two corresponding values of θ between 00 and 3600 for each of them except at the quarter turns, where sin θ and cos θ have values as given in the table below.
Sin θ010-10
Cos θ10-101


  1. (a) Copy and complete the table below giving values of Sin θ correct to 2 decimal places corresponding to θ = 00, 120, 240,        in intervals of 120 up to 3600. Use tables to find Sin θ.
  2. Using scales of 2cm to 600 on the θ axis and 10cm to 1 unit on the Sin θ axis, draw the graph of Sin θ.
  • (a) Given the equation y = sin2θ – cosθ for 00 ≤ θ ≤ 1800, prepare the table of values for the equation (b)Using a scale of 2cm to 300 on the horizontal axis and 5cm to 1 unit on the vertical axis, draw the graph of y= sin2θ – cosθ for 00 ≤ θ ≤ 1800
  • Use your graph to find: (i) the solution of the equation sin2θ – cosθ = 0, correct to the nearest degree.
  • the maximum value of y, correct to 1 d.p

Reading Assignment: NGM for SS 3, Chapter 6, page 46 – 52 Weekend Assignment

  • (a) Draw the graph of the equation y = 1 + cos 2x for 00 ≤ θ ≤ 3600 at interval of 300 Using a scale of 2cm to 300 on the horizontal axis and 2cm to 1 unit on the vertical axis

(b)Use your graph to solve 1 + cos 2x = 0

  • Draw the graph of Sin 3θ for values of θ from 00 to 3600 using the appropriate scales.

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