Application of surds to trigonometrical ratios. Draw the graphs of sine and cosine for angles 00< x < 3600

SUBJECT: MATHEMATICS

CLASS: SS 3

TERM: FIRST TERM

WEEK 3                         DATE………………………

Application of Surds to Trigonometrical Ratios

Sine and Cosine graphs        

• Application of Surds to Trigonometrical Ratios: The following summary shows how to find the sine, cosine and tangent of angles in surd form.
• 1. To find the sine, cosine and tangent of an angle in surd form, use the following formulas:
• • Sine = \(\sqrt{a\cos^2{\theta}+b\sin^2{\theta}}\)
• • Cosine = \(\sqrt{a\sin^2{\theta}+b\cos^2{\theta}}\)
• • Tangent = \(\sqrt{a\sin \theta}{\sqrt{1-b^{2}\sin^2\theta}}\)

Trigonometric ratios of 30⁰ and 60⁰:An equilateral triangle of side 2 units is used to obtain the values of sine, cosine and tangent of angles in surd form.

                                                           A

                                                           60⁰

                                            2                          2

                                  B  60⁰                                            60⁰    C

                                                          2

The triangle is divided into two by drawing a parallel line from A to BC, the resulting triangle is below:

A

30⁰

x                  2

B                     60⁰    C

1

Applying the Pythagoras rule: Hyp² = Opp² + Adj²

2²  = 1² + x²

4 – 1 = x²

x = √3

Hence; 30⁰ :         Sin 30⁰ = ½,    Cos 30⁰ = √3/2,   Tan 30⁰ = 1/√3

              60⁰:          Sin 60⁰ = √3/2,   Cos 60⁰ = ½,    Tan 60⁰ = √3

Trigonometric ratios of 45⁰:A right-angled isosceles triangle in ratio 1: √2: 1 is used to obtain the trigonometrical

ratios:                                                                               A

45⁰

1                  m

B                       45⁰ C

1

m² = 1² + 1²

m = √2

Hence: 45⁰:    Sin 45⁰ = 1/√2 or √2/2,    Cos 45⁰ = 1/√2 or √2/2,    Tan 45⁰ = 1

Examples:

Calculate the lengths marked x, y and z and give your in surd form.

A

                         z           x                        y

                          60⁰30⁰

                   B  3cm C                                        D      

To find x; using triangle ABC

Tan 60⁰ = x

3

√3 = x

3

x = 3√3 cm

In triangle ABC; cos 60⁰ = 3/z

1= 3

2   z

z = 6cm

In triangle ADC, sin 30⁰ = x/y

 1 = 3√3

2       y

y = 2 x 3√3

y = 6√3cm

Evaluation:

Given the figure below                      A

10cm

                                                                                        30⁰45⁰

                                                            B                      C                            D

Calculate (a) |BC|     (b) |CD|   (c) |AD|

  TRIGONOMETRICAL GRAPHS OF SINE AND COSINE OF ANGLES BETWEEN 0⁰< θ < 360⁰

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.

0⁰

90⁰

180⁰

270⁰

360⁰

Sin θ

0

1

0

-1

0

Cos θ

1

0

-1

0

1

Evaluation:

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.

(b)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.

1. (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

(c) Use your graph to find: (i) the solution of the equation sin2θ – cosθ = 0, correct to the nearest degree.

(d) the maximum value of y, correct to 1 d.p

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

Weekend Assignment

1. (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

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

3. Given that cos θ = -0.5x, find the value of sin 3600θ

4. Find the value of (i) cos 1200θ and cos 1800θ

(ii) sin 9 θ

(iii) sin 2124π + 24π/7

5. Find the exact value of cos (π/4 – π/6)

6. A building is 50m high and casts a shadow 20m long on horizontal ground at the same time as an adjacent tree casts a shadow 60m long. How tall is the tree?

7. Evaluate tan10π + 2π/3 correct to 1 decimal place.

8. The area of the shaded region in the figure below is 0.1325 sq units. Calculate its perimeter, correct to 1 decimal place

Evaluation:

1. Copy and complete the table below, giving values of cos θ corresponding to θ = 00, 120, 240, …… in intervals of 30 up to 3600. Use tables to find cos θ values.

2. Using a scale of 2cm to 300 on the θ axis and 1cm to 1 unit on the Cos θ axis draw the graph of Cos θ for 00 ≤ θ ≤ 3600

3. Given the equation y = cos 2x – sin x for 00 ≤ θ ≤ 1800, prepare the table of values for the equation

4. Using a scale of 2cm to 300 on the horizontal axis and 3cm to 1 unit on the vertical axis, draw the graph of y= cos2x -sin x for 00 ≤ θ ≤ 1800

5. Use your graph to find the solution of the equation cos2x – sin x = 0, correct to 3 decimal places

6. Find the exact value of sin (π/4 + π/3)

7. Evaluate tan(π – 18π/7), correct to 1 decimal place.

8. The length of a rectangular swimming pool is given by L = 20 + 3cos θ where 0 ≤ θ ≤ 2π. Find the dimensions of the pool in meters, correct to 1 decimal place.