Download - Trigonometry (1)
14
14.1 Introduction to Trigonometry 14.2 Trigonometry Ratios of Arbitrary Angles14.3 Finding Trigonometric Ratios Without Using
a Calculator
Chapter Summary
Case Study
Trigonometry (1)
14.4 Trigonometric Identities14.5 Trigonometric Equations14.6 Graphs of Trigonometric Functions14.7 Graphical Solutions of Trigonometric Equations
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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 StudyCase 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 canbe displayed by using a CRO.
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In the figure, the centre of the circle is O and its radius is r.
1144 .1 .1 Introduction to TrigonometryIntroduction to Trigonometry
A. A. Angles of RotationAngles 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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Remarks: 1. The figure shows the measures of two different angles: 1
30 and 230.However, they have the same initial side OA and terminal side OP.
1144 .1 .1 Introduction to TrigonometryIntroduction to Trigonometry
A. A. Angles of RotationAngles 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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In a rectangular coordinate plane, the x-axis and the y-axis divide the plane into four parts as shown in the figure.
1144 .1 .1 Introduction to TrigonometryIntroduction to Trigonometry
BB. . QuadrantsQuadrants
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 90Quadrant II: 90 180 Quadrant III: 180
270Quadrant IV: 270
360 Notes: 0, 90, 180 and 270 do not belong to any quadrant.
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For an acute angle , the trigonometric ratios between two sides of a right-angled triangle are
1144 ..22 Trigonometric Ratios of ArbitraryTrigonometric Ratios of Arbitrary AnglesAngles
AA. . DefinitionDefinition
.sideadjacent
side oppositetan
x
y
and hypotenuse
sideadjacent cos
r
x,hypotenuse
side oppositesin
r
y
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.
. have We 22 yxr We can then define the trigonometric ratios of in terms of x, y and r:
x
y
r
x
r
y tanand cos,sin
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For example:In the figure, P(–3 , 4) is a point on the terminal side of the angle of rotation .
1144 ..22 Trigonometric Ratios of ArbitraryTrigonometric Ratios of Arbitrary AnglesAngles
AA. . DefinitionDefinition
We have x 3 and y 4.
.54)3( 22 r
5
4sin
r
y3
4tan
x
y5
3cos
r
x
Now, we can extend the definition for angles greater than 90.
By definition:
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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.
1144 ..22 Trigonometric Ratios of ArbitraryTrigonometric Ratios of Arbitrary AnglesAngles
BB. . Signs of Trigonometric RatiosSigns 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 tanSign of cos Sign of sin Sign ofy-coordinate
Sign ofx-coordinateQuadrant
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1144 ..22 Trigonometric Ratios of ArbitraryTrigonometric Ratios of Arbitrary AnglesAngles
BB. . Signs of Trigonometric RatiosSigns 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 tanSign of cos Sign of sin Sign ofy-coordinate
Sign ofx-coordinateQuadrant
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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We can find the trigonometric ratios of given angles by using a calculator.
1144 ..22 Trigonometric Ratios of ArbitraryTrigonometric Ratios of Arbitrary AnglesAngles
CC. . Using a Calculator to Find TrigonometricUsing a Calculator to Find Trigonometric RatiosRatios
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.)
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1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator
AA. . Angles Formed by Coordinates AxesAngles Formed by Coordinates Axes
Thus, x 0 and y r.
090tan
r
x
y
00
90cos rr
x
190sin r
r
r
y
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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1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator
AA. . Angles Formed by Coordinates AxesAngles Formed by Coordinates Axes
010(r, 0)360
undefined01(0, r)270
010(r, 0)180
undefined01(0, r)90
010(r, 0)0
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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1. Reference Angle
1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator
BB. . By Considering the Reference AnglesBy 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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2. Finding Trigonometric Ratios
1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator
BB. . By Considering the Reference AnglesBy 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 theASTC diagram
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1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator
BB. . By Considering the Reference AnglesBy Considering the Reference Angles
For example, to find tan 240 and cos 240:
3
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
2
1
cosine ratio: ve
360tan 2
160cos
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1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator
CC. . Finding Trigonometric Ratios by AnotherFinding Trigonometric Ratios by Another Given Trigonometric RatioGiven 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.
22 yxr
In the last section, we learnt that the trigonometric ratios can be defined as
,tanandcos,sinx
y
r
x
r
y
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Example 14.1T
Solution:
1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator
If , where 270 360, find the values of
sin and cos . 12
5tan
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.
5,12 yx
P(12, 5) is a point on the terminal side of .
22 yxr r
ysin
13
513
)5(12 22
r
xcos
13
12
CC. . Finding Trigonometric Ratios by AnotherFinding Trigonometric Ratios by Another Given Trigonometric RatioGiven Trigonometric Ratio
By definition,
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Example 14.2T
1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator
222 5)2( x
212 x(rejected)21or 21x
r
x cos5
21
x
ytan
21
2
CC. . Finding Trigonometric Ratios by AnotherFinding Trigonometric Ratios by Another Given Trigonometric RatioGiven Trigonometric Ratio
If , where 180 270, find the values of
cos and tan . 5
2sin
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, thex-coordinate of P must be negative.
21
2
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1144 ..44 Trigonometric IdentitiesTrigonometric 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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1144 ..44 Trigonometric IdentitiesTrigonometric 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 ) tan
1
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Example 14.3T
Solution:
1144 ..44 Trigonometric IdentitiesTrigonometric Identities
Simplify the following expressions. (a) tan (180 ) sin (90 )
cos
)180(cos(b)
sin
costan
)(coscos
sin
cos
)180(cos(b)
cos
cos
1
(a) tan (180 ) sin (90 )
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Example 14.4T
1144 ..44 Trigonometric IdentitiesTrigonometric Identities
cos)180sin(2)90cos()90sin( cossin2))90(180cos())90(180sin(
cossin2)sin(cos cossin2cossin
cossin3
cossin2)]90cos()[90sin(
Simplify sin (90 ) cos (90 ) 2sin (180 ) cos .
Solution:
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Example 14.5T
1144 ..44 Trigonometric IdentitiesTrigonometric Identities
.cos)180(tan
)270sin(cos
1
thatProve2
)180(tan
)270sin(cos
1
L.H.S.2
2tan
))90(180sin(cos
1
2tan
)90sin(cos
1
2tan
coscos
1
2
2
2
cos
sincos
cos1
2
22
sin
cos
cos
sin
cos
Solution:
R.H.S.
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1144 ..55 Trigonometric EquationsTrigonometric Equations
A. Finding Angles from Given TrigonometricA. Finding Angles from Given Trigonometric RatiosRatios
Given that , where 0 360. 2
3sin
Step 1: Since sin 0, may lie in either quadrant III or quadrant IV.
Step 2: Let be the reference angle of .
2
3sin 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.
300or 240
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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1144 ..55 Trigonometric EquationsTrigonometric Equations
A. Finding Angles from Given TrigonometricA. Finding Angles from Given Trigonometric RatiosRatios
In general, for any given trigonometric ratio, it may correspond to more than one angle.
120
120, 240, …
Finding the trigonometric ratio
2
1cos
2
1cos
Finding the corresponding angles
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1144 ..55 Trigonometric EquationsTrigonometric Equations
B. Simple Trigonometric EquationsB. 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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Example 14.6T
1144 ..55 Trigonometric EquationsTrigonometric Equations
B. Simple Trigonometric EquationsB. Simple Trigonometric Equations
2
(cor. to 1 d. p.)
2sin)12(
12
2sin
Hence, 55.938 or 180 55.938 124.1or 9.55
If ( 1)sin 2, where 0 360, find . (Give the answers correct to 1 decimal place.)
Solution:
By using a calculator, the reference angle 55.938.
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1144 ..55 Trigonometric EquationsTrigonometric Equations
C. Other Trigonometric EquationsC. Other Trigonometric Equations
We now try to solve some harder trigonometric equations.
Equation Technique
2sin 3cos 0 Using trigonometric identity
5sin2 4 0 Taking square root
sin 2sin cos 0 Taking out the common factor
2cos2 3sin 0 Transforming into a quadratic equation
Examples:
2
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Example 14.7T
1144 ..55 Trigonometric EquationsTrigonometric Equations
0cos7sin7 (a) cos7sin7
7
7
cos
sin
1tan 315or 135
4cos
1(b)
2
2cos41
4
1cos2
2
1or
2
1cos
300or 240,120,60
C. Other Trigonometric EquationsC. Other Trigonometric Equations
Solve the following equations for 0 360.
(a) 7sin 7cos 0
Solution:
4cos
1 (b)
2
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Example 14.8T
1144 ..55 Trigonometric EquationsTrigonometric Equations
0costancos2
0coscos
sincos2
0cossincos 0)1(sincos
1sinor 0cos
270or 90
C. Other Trigonometric EquationsC. Other Trigonometric Equations
Solve the equation cos2 tan cos 0 for 0 360.Solution:
Factorize the given expression and apply the fact that if ab 0, thena 0 or b 0.
270or 90 270
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Example 14.9T
1144 ..55 Trigonometric EquationsTrigonometric Equations
01sincos2 2 01sin)sin1(2 2 01sinsin22 2
01sinsin2 2 0)1sin2)(1(sin
2
1sinor 1sin
90 30360or 30180
330or 210 ,90
C. Other Trigonometric EquationsC. Other Trigonometric Equations
Solve the equation 2cos2 sin 1 0 for 0 360.
Solution:
Transform the equation into a quadratic equation with sin as the unknown.
01sinsin2 2
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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.
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
AA. . The Graph of y The Graph of y sin x sin x
x 0 30 60 90 120 150 180
y 0 0.5 0.87 1 0.87 0.5 0
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 1 0.87 0.5 0
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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.
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
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.
AA. . The Graph of y The Graph of y sin x sin x
P. 34
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
BB. . The Graph of y The Graph of y cos x 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 0 30 60 90 120 150 180 210 240 270 300 330 360
y 1 0.87 0.5 0 0.5 0.87 1 0.87 0.5 0 0.5 0.87 1
From the above table, we can plot the points on the coordinate plane.
P. 35
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
BB. . The Graph of y The Graph of y cos x 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.
P. 36
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
CC. . The Graph of y The Graph of y tan x 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 0 30 45 60 75 90 105 120 135 150
y 0 0.58 1 1.73 3.73 Undefined 3.73 1.37 1 0.58
x 180 210 225 240 255 270 285 300 315 330 360
y 0 0.58 1 1.73 3.73 Undefined 3.73 1.37 1 0.58 0
The value of y is not defined when x 90 and 270.
P. 37
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
CC. . The Graph of y The Graph of y tan x tan x
The graph of y tan x is drawn as below.x 0 30 45 60 75 90 105 120 135 150
y 0 0.58 1 1.73 3.73 Undefined 3.73 1.37 1 0.58
x 180 210 225 240 255 270 285 300 315 330 360
y 0 0.58 1 1.73 3.73 Undefined 3.73 1.37 1 0.58 0
P. 38
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
CC. . The Graph of y The Graph of y tan x 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, ... .
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.
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
1 cos x 1 4 4cos x 4
4 3 3 4cos x 4 31 3 4cos x 7
The maximum and minimum values are 7 and 1 respectively.
CC. . The Graph of y The Graph of y tan x tan x
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.
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
DD. . Transformation on the Graphs of Transformation on the Graphs of Trigonometric FunctionsTrigonometric Functions
P. 41
Example 14.10T
DD. . Transformation on the Graphs of Transformation on the Graphs of Trigonometric FunctionsTrigonometric 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:
1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions
(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
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.
1144 ..77 Graphical Solutions of Graphical Solutions of Trigonometric EquationsTrigonometric Equations
P. 43
Example 14.11T
Solution:
1144 ..77 Graphical Solutions of Graphical Solutions of Trigonometric EquationsTrigonometric 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
P. 44
Example 14.12T
1144 ..77 Graphical Solutions of Graphical Solutions of Trigonometric EquationsTrigonometric 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
P. 45
In a rectangular coordinate plane, the x-axis and the y-axis divide the plane into four quadrants.
Chapter Chapter SummarySummary14.1 Introduction to Trigonometry
P. 46
Chapter Chapter SummarySummary
x
ytan
r
xcos
r
ysin
The signs of different trigonometric ratios in different quadrants can be memorized by the ASTC diagram.
14.2 Trigonometric Ratios of Arbitrary Angles
P. 47
If is the reference angle of an angle , then sin sin ,
cos cos , tan tan ,
where the choice of the sign ( or ) depends on the quadrant in which lies.
Chapter Chapter SummarySummary14.3 Finding Trigonometric Ratios Without
Using a Calculator
P. 48
Chapter Chapter SummarySummary
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
P. 49
Trigonometric equations can be solved by the algebraic method.
Chapter Chapter SummarySummary14.5 Trigonometric Equations
P. 50
1. Graph of y sin x
Chapter Chapter SummarySummary
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 1.
14.6 Graphs of Trigonometric Functions
P. 51
Trigonometric equations can be solved by the graphical method.
Chapter Chapter SummarySummary14.7 Graphical Solutions of Trigonometric Equations