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

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P. 2 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. Case Study The graph repeats itself at regular intervals. Such an interval is called the period. 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.

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P. 3 In the figure, the centre of the circle is O and its radius is r Introduction to Trigonometry A. Angles of Rotation Suppose OA is rotated about O and it reaches OP, the angle 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 is positive. If OA is rotated in a clockwise direction, then the value of is negative.

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P. 4 Remarks: 1.The figure shows the measures of two different angles: 130 and 230 . However, they have the same initial side OA and terminal side OP Introduction to Trigonometry A. Angles of Rotation 2.The initial side and terminal side of 410 coincide with that of 50 as shown in the figure.

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P. 5 In a rectangular coordinate plane, the x-axis and the y-axis divide the plane into four parts as shown in the figure Introduction to Trigonometry B. Quadrants 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 , Quadrant I:0 90 Quadrant II:90 180 Quadrant III:180 270 Quadrant IV:270 360 Notes: 0 , 90 , 180 and 270 do not belong to any quadrant.

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P. 6 For an acute angle , the trigonometric ratios between two sides of a right-angled triangle are 14.2 Trigonometric Ratios of Arbitrary Angles Angles A. Definition We now introduce a rectangular coordinate plane onto OPQ 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 in terms of x, y and r:

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

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P. 8 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 Trigonometric Ratios of Arbitrary Angles Angles B. Signs of Trigonometric Ratios 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 lies. IV III II I Sign of tan Sign of cos Sign of sin Sign of y-coordinate Sign of x-coordinate Quadrant

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

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

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P Finding Trigonometric Ratios Without Using a Calculator Without Using a Calculator A. Angles Formed by Coordinates Axes Thus, x 0 and y r. 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)., which is undefined.

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P Finding Trigonometric Ratios Without Using a Calculator Without Using a Calculator A. Angles Formed by Coordinates Axes 010(r, 0) 360 undefined0 11(0, r)270 0 11 0 ( r, 0)180 undefined01(0, r) 90 010(r, 0) 00 tan cos sin Coordinates of P Notes: The terminal sides OP of 0 and 360 lie in the same position. Thus, their trigonometric ratios must be the same. Suppose we rotate the terminal side OP through 90 , 180 , 270 and 360 in an anti-clockwise direction.

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P Reference Angle 14.3 Finding Trigonometric Ratios Without Using a Calculator Without Using a Calculator B. By Considering the Reference Angles For each angle of rotation (except for 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 angle . Examples: 140 180 140 40 310 360 310 50 30 30 250 250 180 70

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

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P Finding Trigonometric Ratios Without Using a Calculator 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: Step 2:Determine the sign of the corresponding trigonometric ratio: 240 lies in quadrant III. In quadrant III:tangent ratio: ve tan tan cos cos Step 3:Find the trigonometric ratio of its reference angle : 240 180 60 Step 4:Find the trigonometric ratio of the angle 240 : tan 240 tan 60 cos 240 cos 60 cosine ratio: ve

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P Finding Trigonometric Ratios Without Using a Calculator Without Using a Calculator C. Finding Trigonometric Ratios by Another Given Trigonometric Ratio Given Trigonometric Ratio where P(x, y) is a point on the terminal side of the angle of rotation 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. In the last section, we learnt that the trigonometric ratios can be defined as

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P. 17 Example 14.1T Solution: 14.3 Finding Trigonometric Ratios Without Using a Calculator Without Using a Calculator If, where 270 360 , find the values of sin and cos . 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 . C. Finding Trigonometric Ratios by Another Given Trigonometric Ratio Given Trigonometric Ratio By definition,

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P. 18 Example 14.2T 14.3 Finding Trigonometric Ratios Without Using a Calculator Without Using a Calculator C. Finding Trigonometric Ratios by Another Given Trigonometric Ratio Given Trigonometric Ratio If, where 180 270 , find the values of cos and tan . 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. Since lies in quadrant III, the x-coordinate of P must be negative.

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

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

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

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P. 22 Example 14.4T 14.4 Trigonometric Identities Simplify sin (90 ) cos (90 ) 2sin (180 ) cos . Solution:

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P. 23 Example 14.5T 14.4 Trigonometric Identities Solution:

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P Trigonometric Equations A. Finding Angles from Given Trigonometric Ratios Ratios Given that, where 0 360 . Step 1:Since sin 0, may lie in either quadrant III or quadrant IV. Step 2:Let be the reference angle of . 60 Step 3:Locate the angle and its reference angle in each possible quadrant. Step 4:Hence, if lies in quadrant III, 180 60 240 . If lies in quadrant IV, 360 60 300 . 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:

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

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P Trigonometric Equations B. Simple Trigonometric Equations An equation involving trigonometric ratios of an unknown angle is called a trigonometric equation. Usually, there are certain values of 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 b, a cos b and a tan b, where a and b are real numbers.

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P. 27 Example 14.6T 14.5 Trigonometric Equations B. Simple Trigonometric Equations (cor. to 1 d. p.) Hence, or 180 If ( 1)sin 2, where 0 360 , find . (Give the answers correct to 1 decimal place.) Solution: By using a calculator, the reference angle .

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P Trigonometric Equations C. Other Trigonometric Equations We now try to solve some harder trigonometric equations. EquationTechnique 2sin 3cos 0 Using trigonometric identity 5sin 2 4 0 Taking square root sin 2sin cos 0 Taking out the common factor 2cos 2 3sin 0 Transforming into a quadratic equation Examples:

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P. 29 Example 14.7T 14.5 Trigonometric Equations C. Other Trigonometric Equations Solve the following equations for 0 360 . (a)7sin 7cos 0 Solution:

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P. 30 Example 14.8T 14.5 Trigonometric Equations C. Other Trigonometric Equations Solve the equation cos 2 tan cos 0 for 0 360 . Solution: Factorize the given expression and apply the fact that if ab 0, then a 0 or b 0.

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P. 31 Example 14.9T 14.5 Trigonometric Equations C. Other Trigonometric Equations Solve the equation 2cos 2 sin 1 0 for 0 360 . Solution: Transform the equation into a quadratic equation with sin as the unknown.

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P. 32 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 . Consider y sin x. For every angle x, there is a corresponding trigonometric ratio y. Thus, y is a function of x Graphs of Trigonometric Functions A. The Graph of y sin x x 00 30 60 90 120 150 180 y From the above table, we can plot the points on the coordinate plane. x 210 240 270 300 330 360 y 0.5 0.87 11 0.5 0

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P. 33 The graph of y sin x repeats itself in the intervals –360 x 0 , 0 x 360 , 360 x 720 , etc. We can also plot the graph of y sin x for 360 x 720 , etc Graphs of Trigonometric Functions 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 . A. The Graph of y sin x

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P Graphs 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. x 00 30 60 90 120 150 180 210 240 270 300 330 360 y 0.5 0.87 11 From the above table, we can plot the points on the coordinate plane.

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P Graphs 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 .

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P Graphs of Trigonometric Functions C. The Graph of y tan x 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. 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. x 00 30 45 60 75 90 105 120 135 150 y Undefined 3.73 1.37 11 0.58 x 180 210 225 240 255 270 285 300 315 330 360 y Undefined 3.73 1.37 11 The value of y is not defined when x 90 and 270 .

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P Graphs of Trigonometric Functions C. The Graph of y tan x The graph of y tan x is drawn as below. x 00 30 45 60 75 90 105 120 135 150 y Undefined 3.73 1.37 11 0.58 x 180 210 225 240 255 270 285 300 315 330 360 y Undefined 3.73 1.37 11

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P Graphs 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: 2.y tan x is a periodic function with a period of 180 . From 0 to 90 , tan x increases from 0 to positive infinity. From 90 to 180 , tan x increases from negative infinity to 0. 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 ,....

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P. 39 For example, to find the maximum and minimum values of 3 4cos x: Given a trigonometric function, we can find its maximum and minimum values algebraically Graphs of Trigonometric Functions 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. C. The Graph of y tan x

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

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P. 41 Example 14.10T D. Transformation on the Graphs of Trigonometric Functions Trigonometric Functions (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 Solution: 14.6 Graphs of Trigonometric Functions (a)Refer to the figure. (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. y cos x 2 y cos (x 180 )y cos x

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

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P. 43 Example 14.11T Solution: 14.7 Graphical Solutions of Trigonometric Equations Trigonometric Equations 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 (a)Draw the straight line y 0.6 on the graph. The straight line cuts the curve at x 54 and 306. So the solution of cos x 0.6 for 0 x 360 is 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 for 0 x 360 is 135 or 225. y 0.6 y 0.7

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P. 44 Example 14.12T 14.7 Graphical Solutions of Trigonometric Equations Trigonometric Equations 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: (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. y 1.5

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

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

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P. 47 If is the reference angle of an angle , thensin sin , cos cos , tan tan , where the choice of the sign ( or ) depends on the quadrant in which lies. Chapter Summary 14.3 Finding Trigonometric Ratios Without Using a Calculator

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

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

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P Graph of y sin x Chapter Summary 2.Graph of y cos x 3.Graph of y tan x 5.The periods of sin x, cos x and tan x are 360 , 360 and 180 respectively. 4. For any real value of x, 1 sin x 1 and 1 cos x Graphs of Trigonometric Functions

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P. 51 Trigonometric equations can be solved by the graphical method. Chapter Summary 14.7 Graphical Solutions of Trigonometric Equations

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Follow-up Finding Trigonometric Ratios Without Using a Calculator Without Using a Calculator C. Finding Trigonometric Ratios by Another Given Trigonometric Ratio Given Trigonometric Ratio Solution: If, where 180 270 , find the values of sin and cos . Since tan 0, lies in quadrant I or III. As it is given that 180 270 , must lie in quadrant III where sin 0 and cos 0. P( 4, 3) is a point on the terminal side of . By definition,

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Follow-up Finding Trigonometric Ratios Without Using a Calculator Without Using a Calculator C. Finding Trigonometric Ratios by Another Given Trigonometric Ratio Given Trigonometric Ratio If, where 90 180 , find the values of sin and tan . Since cos 0 and 90 180 , lies in quadrant II. Let P( 3, y) be a point on the terminal side of . We have x 3 and r 5. Since lies in quadrant II, the y-coordinate of P must be positive. Solution:

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Follow-up Trigonometric Identities Simplify the following expressions. (a)cos (180 ) tan (180 ) Solution:

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Follow-up Trigonometric Identities Simplify tan (90 ) sin (180 ) 4cos (180 ). Solution:

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Follow-up Trigonometric Identities Solution:

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Follow-up Trigonometric Equations B. Simple Trigonometric Equations (cor. to 1 d. p.) Hence, 180 or 180 Solve 5cos 2, where 0 360 . (Give the answers correct to 1 decimal place.) Solution: By using a calculator, the reference angle .

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Follow-up 14.7 Solution: 14.5 Trigonometric Equations (cor. to 1 d. p.) C. Other Trigonometric Equations Solve the following equations for 0 360 . (Give the answers correct to 1 decimal place.) (a)6sin 8cos 0(b)tan 2 2 0

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Follow-up Trigonometric Equations C. Other Trigonometric Equations Solve the equation tan 2sin 0 for 0 360 . Solution: As cos 0, we can multiply the equation by cos .

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Follow-up Trigonometric Equations C. Other Trigonometric Equations Solve the equation 2sin 2 3cos 0 for 0 360 . Solution: Note that 1 cos 1. For details, refer to the graph of y cos x in Section 14.6.

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Follow-up Graphs of Trigonometric Functions D. Transformation on the Graphs of Trigonometric Functions Trigonometric Functions The following figure shows the graph of the function y tan x for 90 x 540 . Sketch the graphs of the following functions on the figure. (i)y tan x 1 (ii)y tan x Solution: (i)The graph of the function y tan x 1 is obtained by translating the graph of y tan x one unit downwards. (ii)The graph of the function y tan x is obtained by reflecting the graph of y tan x about the x-axis. y tan x 1 y tan x

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Follow-up Graphical Solutions of Trigonometric Equations Trigonometric Equations Consider the graph of y tan x for 0 x 360. Using the graph, solve the following equations. (a)tan x 3 (b)tan x 2 (a)Draw the straight line y 3 on the graph. The straight line cuts the curve at x 72 and 252. So the solution of tan x 3 for 0 x 360 is 72 or 252. (b)Draw the straight line y 2 on the graph. The straight line cuts the curve at x 117 and 297. So the solution of tan x 2 for 0 x 360 is 117 or 297. y 3 y 2y 2 Solution:

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Follow-up Graphical Solutions of Trigonometric Equations Trigonometric Equations The figure shows the graph of y acos x bsin x for 0 x 360. (a)Find the values of a and b. (b)Using the graph, solve the equation 2cos x 3sin x 2. Solution: (a)Since the graph passes through (0, 2), we have Since the graph passes through (90, 3), we have (b)Draw the straight line y 2 on the graph. The straight line cuts the curve at x 66 and 180. Therefore, the solution is 66 or 180. y 2

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