Chapter 7 Trigonometry Additional Example 7.1Additional Example 7.1 Additional Example 7.2Additional Example 7.2 Additional Example 7.3Additional Example 7.3 Additional Example 7.4Additional Example 7.4 Additional Example 7.5Additional Example 7.5 Additional Example 7.6Additional Example 7.6 Additional Example 7.7Additional Example 7.7 Additional Example 7.8Additional Example 7.8 Additional Example 7.9Additional Example 7.9 Additional Example 7.10Additional Example 7.10 Example 1Example 1 Example 2Example 2 Example 3Example 3 Example 4Example 4 Example 5Example 5 Example 6Example 6 Example 7Example 7 Example 8Example 8 Example 9Example 9 Example 10Example 10 New Trend Mathematics - S4B Quit

Chapter 7 Trigonometry Additional Example 7.11Additional Example 7.11 Additional Example 7.12Additional Example 7.12 Additional Example 7.13Additional Example 7.13 Additional Example 7.14Additional Example 7.14 Additional Example 7.15Additional Example 7.15 Additional Example 7.16Additional Example 7.16 Additional Example 7.17Additional Example 7.17 Additional Example 7.18Additional Example 7.18 Additional Example 7.19Additional Example 7.19 Additional Example 7.20Additional Example 7.20 Example 11Example 11 Example 12Example 12 Example 13Example 13 Example 14Example 14 Example 15Example 15 Example 16Example 16 Example 17Example 17 Example 18Example 18 Example 19Example 19 Example 20Example 20 New Trend Mathematics - S4B Quit

Chapter 7 Trigonometry Additional Example 7.21Additional Example 7.21 Additional Example 7.22Additional Example 7.22 Additional Example 7.23Additional Example 7.23 Additional Example 7.24Additional Example 7.24 Additional Example 7.25Additional Example 7.25 Example 21Example 21 Example 22Example 22 Example 23Example 23 Example 24Example 24 Example 25Example 25 New Trend Mathematics - S4B Quit

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.1 Solution: In the figure, BCD is a straight line. Find x and . (Correct your answers to 1 decimal place.)

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.2 If tan  = 2 where 0  <  < 90 , find sin  and cos . Solution: By the Pythagoras’ theorem, [Construct  ABC such that BC  2, AC  1 and  ACB  90 .]

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.3 Solution: Find the value of each of the following expressions.

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.4 In the figure, find sin , cos  and tan . Solution: According to the Pythagoras’ theorem,

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.5 Express the following trigonometric ratios in terms of acute angles. (a)sin 217  (b)cos 343  (c)tan 122  Solution: Reference angle  37  Reference angle  17  Reference angle  58 

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.6 Find the values of the following trigonometric ratios. (a)cos 120  (b)sin 315  (c)tan 330  Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.7 Let A be a point on the terminal side of  which lies in quadrant II. It is given that the y-coordinate of A is 3 and r = 7. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.8 If tan    4 where 270  <  < 360 , find sin  and cos . Let  be the reference angle of . Consider a right-angled triangle ABC with tan   4. By the Pythagoras’ theorem, Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.9 By the Pythagoras’ theorem, Solution:  cos  is negative.  lies in quadrant II or quadrant III. If  lies in quadrant II, Let  be the reference angle of . Consider a right-angled triangle ABC with

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.9 If  lies in quadrant III,

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.10 Simplify the following expressions. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.11 Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.12 Proof:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.13 According to the graph in Example 13, find the range of x such that sin x <  0.7. From the graph, when sin x <  0.7, the range of x is Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.14 Sketch the graphs of the following functions on the graph of (b)The graph of y = 2cos x  1 can be obtained by transforming the original graph according to the following steps. Solution: (a)The graph of y = cos(x + 60  ) can be obtained by shifting the original graph 60  to the left.

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.14 Step 2:Shift the new graph obtained in step 1 one unit downwards to get the graph of y = 2 cos x  1. Solution: Step 1:Double the amplitude to get the graph of y = 2 cos x.

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.14 Step 2:Shift the new graph obtained in step 1 two units upwards to get the graph (c)The graph of can be obtained by transforming the original graph according to the following steps. Solution: Step 1:Double the period to get the graph of

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.15 The figure shows the graph of (a)From the graph of (i)the maximum and minimum values. (ii)the period. (iii)the amplitude. (b)Find x such that y attains its maximum value. (c)Find the values of a and b. Solution: (a)From the graph, (i)maximum value; minimum value (ii)period (iii)amplitude (b)From the graph, when y attains its maximum value,

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.15 Substitute b = 3 into (2), Solution: When x  240 , y  5.

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.16 Find the maximum and minimum values of the following expressions for 0   x  360 . (a)  The maximum and minimum values of cos x are 1 and  1 respectively. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.16 (b)  The maximum and minimum values of sin x are 1 and  1 respectively. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.16 (c)  The maximum and minimum values of cos 2 x are 1 and 0 respectively. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.17 Solve the following equations for 0   x  360 . (a)[Since sin 50  > 0, we take 50  as the reference angle.] When sin (x  40  ) > 0, (x  40  ) lies in quadrant I or II. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit (c) Additional Example 7.17 (b)[Since tan 45   1, we take 45  as the reference angle.] When tan (x + 70  ) < 0, (x + 70  ) lies in quadrant II or IV. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.18 Solve 2 cos x + 7 sin x = 0 for 0   x  360 , correct your answers to 1 decimal place. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.19 Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.20 Solve 3sin 2   2sin   1  0 for 0     360 , correct your answers to 1 decimal place if necessary. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.21 Solve 4cos 2   sin   3 for 0     360 , correct your answers to 1 decimal place. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.22 Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.23 Solve correct your answers to 1 decimal place. Solution:

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 7.24 From the graph, when y = 0, Solution: The figure shows the graph of y  sin 2x  cos 2x for 0   x  180 . Solve the equation sin 2x  cos 2x  0 for 0   x  180  graphically.

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 7 Trigonometry 2004 Chung Tai Educational Press © Quit From the graph, when y  4 or y   2, the equation sin 3 x  3cos x  k  0 does not have any solution for 0   x  360 . Solution: Additional Example 7.25 According to the graph in Example 25, find the range of positive values of k such that sin 3 x  3cos x  k  0 does not have any solution for 0     360 .