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Trigonometry (1)

1 4. Trigonometry (1). Case Study. 1 4 .1 Introduction to Trigonometry. 1 4 .2 Trigonometry Ratios of Arbitrary Angles. 1 4 .3 Finding Trigonometric Ratios Without Using a Calculator. 1 4 . 4 Trigonometric Identities. 1 4 . 5 Trigonometric Equations.

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Trigonometry (1)

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  1. 14 Trigonometry (1) Case Study 14.1Introduction to Trigonometry 14.2 Trigonometry Ratios of Arbitrary Angles 14.3 Finding Trigonometric Ratios Without Using a Calculator 14.4Trigonometric Identities 14.5Trigonometric Equations 14.6Graphs of Trigonometric Functions 14.7Graphical Solutions of Trigonometric Equations Chapter Summary

  2. How can we find the shape of the sound wave generated by a tuning fork? The sound wave generated can be displayed by using a CRO. Case Study The figure shows the sound wave generated by the tuning fork displayed on a cathode-ray oscilloscope (CRO). The pattern of the waveform of sound has the same shape as the graph of a trigonometric function. The graph repeats itself at regular intervals. Such an interval is called the period.

  3. 14.1 Introduction to Trigonometry A. Angles of Rotation In the figure, the centre of the circle is O and its radius is r. Suppose OA is rotated about O and it reaches OP, the angle q formed is called an angle of rotation. • OA: initial side • OP: terminal side If OA is rotated in an anti-clockwise direction, the value of qis positive. If OA is rotated in a clockwise direction, then the value of q is negative.

  4. 14.1 Introduction to Trigonometry A. Angles of Rotation • Remarks: • The figure shows the measures of two different angles: 130 and 230. • However, they have the same initial side OA and terminal side OP. 2. The initial side and terminal side of 410 coincide with that of 50 as shown in the figure.

  5. 14.1 Introduction to Trigonometry B. Quadrants In a rectangular coordinate plane, the x-axis and the y-axis divide the plane into four parts as shown in the figure. Each part is called a quadrant. Notes: The x-axis and the y-axis do not belong to any of the four quadrants. For an angle of rotation, the position where the terminal side lies determines the quadrant in which the angle lies. Thus, we can see that for an angle of rotation q, • Quadrant I: 0q 90 • Quadrant II: 90q 180 • Quadrant III: 180q 270 • Quadrant IV: 270q 360 Notes: 0, 90, 180 and 270 do not belong to any quadrant.

  6. 14.2Trigonometric Ratios of Arbitrary Angles A. Definition For an acute angle q, the trigonometric ratios between two sides of a right-angled triangle are We now introduce a rectangular coordinate plane onto DOPQ such that OP is the terminal side as shown in the figure. Suppose the coordinates of P are (x, y) and the length of OP is r. We can then define the trigonometric ratios of q in terms of x, y and r:

  7. 14.2Trigonometric Ratios of Arbitrary Angles A. Definition Now, we can extend the definition for angles greater than 90. For example: In the figure, P(–3, 4) is a point on the terminal side of the angle of rotation q. We have x 3 and y 4. By definition:

  8. Quadrant Sign of x-coordinate Sign of y-coordinate Sign of sin q Sign of cos q Sign of tan q 14.2Trigonometric Ratios of Arbitrary Angles B. Signs of Trigonometric Ratios In the previous section, we defined the trigonometric ratios in terms of the coordinates of a point P(x, y) on the terminal side and the length r of OP. Since x and y may be either positive or negative, the trigonometric ratios may be either positive or negative depending upon the quadrant in which q lies. I      II      III      IV     

  9. Quadrant Sign of x-coordinate Sign of y-coordinate Sign of sin q Sign of cos q Sign of tan q I      II      III      IV      14.2Trigonometric Ratios of Arbitrary Angles B. Signs of Trigonometric Ratios The signs of the three trigonometric ratios in different quadrants can be summarized in the following diagram which is called an ASTC diagram. A : All positive S : Sine positive T : Tangent positive C : Cosine positive Notes: ‘ASTC’ can be memorized as ‘Add Sugar To Coffee’.

  10. 14.2Trigonometric Ratios of Arbitrary Angles C. Using a Calculator to Find Trigonometric Ratios We can find the trigonometric ratios of given angles by using a calculator. For example, (a) sin 160 0.342 (cor. to 3 sig. fig.) (b) tan 245 2.14 (cor. to 3 sig. fig.) (c) cos(123)0.545 (cor. to 3 sig. fig.) (d) sin(246) 0.914 (cor. to 3 sig. fig.)

  11. 14.3Finding Trigonometric Ratios Without Using a Calculator A. Angles Formed by Coordinates Axes If we rotate the terminal side OP with length r units (r 0) through 90 in an anti-clockwise direction, then the coordinates of P are (0, r). Thus, x 0 and yr. , which is undefined.

  12. q Coordinates of P sin q cos q tan q 0 (r, 0) 0 1 0 90 (0, r) 1 0 undefined 14.3Finding Trigonometric Ratios Without Using a Calculator A. Angles Formed by Coordinates Axes Suppose we rotate the terminal side OP through 90, 180, 270 and 360 in an anti-clockwise direction. 180 (r, 0) 0 1 0 270 (0, r) 1 0 undefined 360 (r, 0) 0 1 0 Notes: The terminal sides OP of q 0 and 360 lie in the same position. Thus, their trigonometric ratios must be the same.

  13. 14.3Finding Trigonometric Ratios Without Using a Calculator B. By Considering the Reference Angles 1. Reference Angle For each angle of rotation q (except for q 90n, where n is an integer), we consider the corresponding acute angle measured between the terminal side and the x-axis. It is called the reference angleb. Examples: q 30  b 30 q 140  b 180 140  40 q 250  b 250 180  70 q 310  b 360 310  50

  14. According to the ASTC diagram 14.3Finding Trigonometric Ratios Without Using a Calculator B. By Considering the Reference Angles 2. Finding Trigonometric Ratios By using the reference angle, we can find the trigonometric ratios of an arbitrary angle. The following four steps can help us find the trigonometric ratio of any given angle q: Step 1: Determine the quadrant in which the angle q lies. Step 2: Determine the sign of the corresponding trigonometric ratio. Step 3: Find the trigonometric ratio of its reference angle b. Step 4: Find the trigonometric ratio of the angle q by assigning the sign determined in step 2 to the ratio determined in step 3.

  15. 14.3Finding Trigonometric Ratios Without Using a Calculator B. By Considering the Reference Angles For example, to find tan 240 and cos 240: Step 1: Determine the quadrant in which the angle 240 lies:  240 lies in quadrant III. Step 2: Determine the sign of the corresponding trigonometric ratio:  In quadrant III: tangent ratio: ve cosine ratio: ve \ tan q tan b cos qcos b Step 3: Find the trigonometric ratio of its reference angle b:  b 240 180  60 Step 4: Find the trigonometric ratio of the angle 240:  tan 240 tan 60 cos 240cos 60

  16. 14.3Finding Trigonometric Ratios Without Using a Calculator C. Finding Trigonometric Ratios by Another Given Trigonometric Ratio In the last section, we learnt that the trigonometric ratios can be defined as where P(x, y) is a point on the terminal side of the angle of rotation q and is the length of OP. Now, we can use the above definitions to find other trigonometric ratios of an angle when one of the trigonometric ratios is given.

  17. 14.3Finding Trigonometric Ratios Without Using a Calculator C. Finding Trigonometric Ratios by Another Given Trigonometric Ratio Example 14.1T If , where 270q 360, find the values of sin q and cos q. Solution: Since tan  0,  lies in quadrant II or IV. As it is given that 270 360,  must lie in quadrant IV where sin  0 and cos  0. P(12, 5) is a point on the terminal side of . By definition,

  18. Since  lies in quadrant III, the x-coordinate of P must be negative. 14.3Finding Trigonometric Ratios Without Using a Calculator C. Finding Trigonometric Ratios by Another Given Trigonometric Ratio Example 14.2T If , where 180q 270, find the values of cos q and tan q. Solution: Since sin  0 and 180 270, lies in quadrant III. Let P(x, 2) be a point on the terminal side of . We have y2 and r  5.

  19. 14.4Trigonometric Identities With the help of reference angles in the last section, we can get the following important identities. For any acute angle q, since 180  q lies in quadrant II, we have sin (180 q)sin q cos (180 q)cos q tan (180 q)tan q Since 180q lies in quadrant III, we have sin (180q)sin q cos (180q)cos q tan (180q) tan q

  20. 14.4Trigonometric Identities Since 360q lies in quadrant IV, we have sin (360q)sin q cos (360q) cos q tan (360q)tan q Notes: The above identities also hold if q is not an acute angle. They are useful in simplifying expressions involving trigonometric ratios. Remarks: The following identities also hold if q is not an acute angle: sin (90q) cos q cos (90q) sin q tan (90q)

  21. 14.4Trigonometric Identities Example 14.3T Simplify the following expressions. (a) tan (180q) sin (90  q) Solution: (a) tan (180q) sin (90  q)

  22. 14.4Trigonometric Identities Example 14.4T Simplify sin (90q) cos (90q) 2sin (180q) cos q. Solution:

  23. 14.4Trigonometric Identities Example 14.5T Solution:

  24. 14.5Trigonometric Equations A. Finding Angles from Given Trigonometric Ratios In previous sections, we learnt how to find the trigonometric ratios of any angle. Now, we will study how to find the angle if a trigonometric ratio of the angle is given. For example: Given that , where 0q 360. Step 1: Since sin q 0, q may lie in either quadrant III or quadrant IV. Step 2: Let b be the reference angle of q.  b 60 Step 3: Locate the angle q and its reference angle b in each possible quadrant. Step 4: Hence, ifq lies in quadrant III, q 180 60 240. Ifq lies in quadrant IV, q360 60300.

  25. Finding the trigonometric ratio q 120 q 120, 240, … Finding the corresponding angles 14.5Trigonometric Equations A. Finding Angles from Given Trigonometric Ratios In general, for any given trigonometric ratio, it may correspond to more than one angle.

  26. 14.5Trigonometric Equations B. Simple Trigonometric Equations An equation involving trigonometric ratios of an unknown angle q is called a trigonometric equation. Usually, there are certain values of q which satisfy the given equation. The process of finding the solutions of the equation is called solving trigonometric equation. We will try to solve some simple trigonometric equations: a sin qb, a cos qb and a tan qb, where a and b are real numbers.

  27. By using a calculator, the reference angle 55.938. 14.5Trigonometric Equations B. Simple Trigonometric Equations Example 14.6T If (  1)sin q 2, where 0q 360, find q. (Give the answers correct to 1 decimal place.) Solution: Hence, 55.938or 180 55.938 (cor. to 1 d. p.)

  28. 14.5Trigonometric Equations C. Other Trigonometric Equations We now try to solve some harder trigonometric equations. Examples:

  29. 14.5Trigonometric Equations C. Other Trigonometric Equations Example 14.7T Solve the following equations for 0 q 360. (a) 7sin q 7cos q 0 Solution:

  30. Factorize the given expression and apply the fact that if ab 0, then a 0 or b 0. 14.5Trigonometric Equations C. Other Trigonometric Equations Example 14.8T Solve the equation cos2q tan q  cos q 0 for 0 q 360. Solution:

  31. Transform the equation into a quadratic equation with sin q as the unknown. 14.5Trigonometric Equations C. Other Trigonometric Equations Example 14.9T Solve the equation 2cos2q sin q 1  0 for 0 q 360. Solution:

  32. 14.6Graphs of Trigonometric Functions A. The Graph of y  sin x Consider y sin x. For every angle x, there is a corresponding trigonometric ratio y. Thus, y is a function of x. The following table shows some values of x and the corresponding values of y (correct to 2 decimal places if necessary) for 0 £x£ 360. From the above table, we can plot the points on the coordinate plane.

  33. 14.6Graphs of Trigonometric Functions A. The Graph of y  sin x We can also plot the graph of y sin x for 360£x£ 720, etc. The graph of y sin xrepeats itself in the intervals –360£x £ 0, 0£x£ 360, 360£x£720, etc. Remarks: A function repeats itself at regular intervals is called a periodic function. The regular interval is called a period. From the figure, we obtain the following results for the graph of y sin x for 0£x£ 360: 1. The domain of y sin x is the set of all real numbers. 2. The maximum value of y is 1, which corresponds to x  90. The minimum value of y is –1, which corresponds to x 270. 3. The function is a periodic function with a period of 360.

  34. 14.6Graphs of Trigonometric Functions B. The Graph of y  cos x The following table shows some values of x and the corresponding values of y (correct to 2 decimal places if necessary) for 0 £x£ 360 for y cos x. From the above table, we can plot the points on the coordinate plane.

  35. 14.6Graphs of Trigonometric Functions B. The Graph of y  cos x From the figure, we obtain the following results for the graph of y cos x for 0£x£ 360: 1. The domain of y cos x is the set of all real numbers. 2. The maximum value of y is 1, which corresponds to x  0 and 360. The minimum value of y is –1, which corresponds to x 180. Notes: If we plot the graph of y cos x for –360 £x£ 720, we can see that the graph repeats itself every 360. Thus, y cos x is a periodic function with a period of 360.

  36. 14.6Graphs of Trigonometric Functions C. The Graph of y  tan x The following table shows some values of x and the corresponding values of y (correct to 2 decimal places if necessary) for 0 £x£ 360 for y tan x. The value of y is not defined when x 90 and 270. When an angle is getting closer and closer to 90 or 270, the corresponding value of tangent function approaches to either positive infinity or negative infinity.

  37. 14.6Graphs of Trigonometric Functions C. The Graph of y  tan x The graph of y tan x is drawn as below.

  38. 14.6Graphs of Trigonometric Functions C. The Graph of y  tan x From the figure, we obtain the following results for the graph of y tan x: 1. For 0 £x£ 180, y tan x exhibits the following behaviours: From 0 to 90, tan x increases from 0 to positive infinity. From 90 to 180, tan x increases from negative infinity to 0. 2. y tan x is a periodic function with a period of 180. 3. As tan x is undefined when x 90 and 270, the domain of y tan x is the set of all real numbers except x  90, 270, ... .

  39. 14.6Graphs of Trigonometric Functions C. The Graph of y  tan x Given a trigonometric function, we can find its maximum and minimum values algebraically. For example, to find the maximum and minimum values of 3  4cos x: 1 cos x 1 4 4cos x 4 4  3 3  4cos x 4  3 1 3  4cos x 7 The maximum and minimum values are 7 and 1 respectively.

  40. 14.6Graphs of Trigonometric Functions D. Transformation on the Graphs of Trigonometric Functions In Book 4, we learnt the transformations such as translation and reflection of graphs of functions. Now, we will study the transformations on the graphs of trigonometric functions.

  41. 14.6Graphs of Trigonometric Functions D. Transformation on the Graphs of Trigonometric Functions Example 14.10T (a) Sketch the graph of y cos x for 180 £x£ 360. (b) From the graph in (a), sketch the graphs of the following functions. (i) y cos x 2 (ii) y cos (x 180) (iii) ycos x ycos x y cos (x 180) Solution: (a) Refer to the figure. y cos x 2 (b) The graph of the function (i) y cos x 2 is obtained by translating the graph of y cos x two units downwards. (ii) y cos (x 180) is obtained by translating the graph of y cos x to the left by 180. (ii) ycos x is obtained by reflecting the graph of y cos x about the x-axis.

  42. 14.7Graphical Solutions of Trigonometric Equations Similar to quadratic equations, trigonometric equations can be solved either by the algebraic method or the graphical method. We should note that the graphical solutions are approximate in nature.

  43. 14.7Graphical Solutions of Trigonometric Equations Example 14.11T Consider the graph of y  cos x for 0 £x£ 360. Using the graph, solve the following equations. (a) cos x 0.6 (b) cos x0.7 y 0.6 Solution: (a) Draw the straight line y 0.6 on the graph. The straight line cuts the curve at x 54 and 306. y0.7 So the solution of cos x 0.6 for0 £x£ 360is 54 or 306. (b) Draw the straight line y0.7 on the graph. The straight line cuts the curve at x 135 and 225. So the solution of cos x0.7 for0 £x£ 360is 135 or 225.

  44. 14.7Graphical Solutions of Trigonometric Equations Example 14.12T Draw the graph of y 3cos x sin x for 0 £x£ 360. Using the graph, solve the following equations for 0 £x£ 360. (a) 3cos x sin x  0 (b) 3cos x sin x  1.5 Solution: y 1.5 (a) From the graph, the curve cuts the x-axis at x 72 and 252. Therefore, the solution is 72 or 252. (b) Draw the straight line y1.5 on the graph. The straight line cuts the curve at x 43 and 280. Therefore, the solution is 43 or 280.

  45. Chapter Summary 14.1 Introduction to Trigonometry In a rectangular coordinate plane, the x-axis and the y-axis divide the plane into four quadrants.

  46. Chapter Summary 14.2 Trigonometric Ratios of Arbitrary Angles The signs of different trigonometric ratios in different quadrants can be memorized by the ASTC diagram.

  47. Chapter Summary 14.3Finding Trigonometric Ratios Without Using a Calculator If b is the reference angle of an angle q, then sin qsin b, cos qcos b, tan qtan b, where the choice of the sign ( or ) depends on the quadrant in which q lies.

  48. Chapter Summary 14.4Trigonometric Identities 1. (a) sin (180– q) sin q (b) cos (180– q)–cos q (c) tan (180– q)–tan q 2. (a) sin (180q)–sin q (b) cos (180q)–cos q (c) tan (180q) tan q 3. (a) sin (360–q)–sin q (b) cos (360–q) cos q (c) tan (360–q)–tan q

  49. Chapter Summary 14.5Trigonometric Equations Trigonometric equations can be solved by the algebraic method.

  50. Chapter Summary 14.6Graphs of Trigonometric Functions 1. Graph of y sin x 2. Graph of ycosx 3. Graph of ytanx 4.For any real value of x, 1  sin x 1 and 1  cos x 1. 5. The periods of sin x, cos x and tan x are 360, 360 and 180respectively.

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