Chapter 8.1: Circular Functions (Trigonometry)
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1 Chapter 8.1: Circular Functions (Trigonometry) SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza
2 Chapter 8.1: Circular Functions Lecture 8.1: Basic Concepts Lecture 8.2: Degree Measure and Degree Radian Measure Lecture 8.3: Standard Angle and Coterminal Angle Lecture 8.4: Circular Functions
3 Lecture 8.1: Basic Concepts SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza
4 Ray and Vertex A ray consists of point O on a line and extends indefinitely in one direction. The point O is called the end point (or vertex).
5 Did you know? If two rays are drawn with a common endpoint, then they will form an angle.
6 Initial Side and Terminal Side The initial position of the ray is called the initial side of the angle, while the position of the ray after rotation is called the terminal side.
7 Initial Side and Terminal Side
8 Take Note: The ray can travel an unlimited number of times around the circle and still end in the same terminal position.
9 Standard Angle An angle is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the x-axis.
10 Standard Angle
11 Quadrantal Angle When the terminal side of an angle in standard position coincides with one of the rays of either the x-axis or the y-axis, the angle is said to be a quadrantal angle.
12 Quadrantal Angle
13 Something to think about How many quadrantal angles do we have?
14 Did you know? There are infinitely many quandrantal angles and its measure is an integral multiple of 90.
15 Positive and Negative Angles Angles formed by a counter clockwise direction are considered positive angles; angles formed by a clockwise direction are considered negative angles.
16 Positive and Negative Angles
17 Lecture 8.2: Degree Measure and Radian Measure SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza
18 Our Goal: This section focuses on the degree measure and radian measure of an angle. This section also includes the different angles and its measure.
19 Measure of an Angle The measure of an angle is given by stating the amount of rotation to revolve from the initial position of the ray to the terminal position.
20 Units to Measure Angles The most commonly used unit to measure angles are in terms of: 1. revolution; 2. degrees; and 3. radians.
21 One Revolution One revolution is the amount of rotation needed for one full turn of a ray about its endpoints in which the initial and terminal sides of the angle coincides.
22 One Degree A degree is the measure of an angle formed by rotation a ray 1/360 of a complete revolution, the symbol for degree is.
23 Types of Angles One Revolution Straight Angle Right Angle Acute Angle Obtuse Angle Reflex Angle
24 Straight Angle A straight angle is an angle of 180, or ½ revolution.
25 Right Angle A right angle is an angle of 90.
26 Acute Angle An acute angle is an angle if its measure is between 0 and 90.
27 Obtuse Angle An obtuse angle is an angle if its measure is between 90 and 180.
28 Reflex Angle A reflex angle is an angle if its measure is between 180 and 360.
29 Complementary Angles If the sum of the two measures of two angles is 90 (or α + β = 90 ) the angles are complementary angles and one angle is the complement of the other.
30 Supplementary Angles If the sum of the two measures of two angles is 180 (or α + β = 180 ) the angles are supplementary angles and one angle is the supplement of the other.
31 Take Note: Complementary and Supplementary angles are ALWAYS positive.
32 Complementary Angle 90 2
33 Supplementary Angle 180
34 Example 8.2.1: Find the complementary and supplementary angles of: 6
35 Final Answer: The complement is 3.
36 Final Answer: The supplement is 5 6.
37 Example 8.2.2: Find the complementary and supplementary angles of: 72
38 Final Answer: The complement of 72 is 18.
39 Final Answer: The supplement of 72 is 108.
40 Example 8.2.3: Find the complementary and supplementary angles of: 2 3
41 Since Final Answer: 2 3 it has no complement. 2,
42 Final Answer: The supplement is: 3
43 Did you know? In geometry you learned to measure angles in degrees. In trigonometry, angles are also measured in radians.
44 Radian Measure To define this unit of measure, consider a circle with center at the origin and radius equal to r, and let θ be an angle in standard position. Let s be the length of the arc intercepted by the angle. If s = r, then the measure of angle is said to be equal to one radian.
45 Radian Measure
46 Radian Measure One radian is equivalent to the measure of a central angle θ that intercepts an arc s = r of the circle.
47 In Other Words One radian is the measure of a central angle that intercepts an arc S equal in length to the radius of the circle.
48 Understanding Radian Measure
49 Take Note: Radians have no units. Therefore, when radians are being used it is customary that no units are indicated for the angle.
50 Take Note: Note that θ is taken in degrees if it is indicated; otherwise θ is in radian measure.
51 Did you know? There is a little more than 6 radius lengths that can be wrapped around one full circle.
52 Radians
53 Did you know? If θ is the measure of an angle in radians, then, s r and θ are related by the equation: s s r r
54 Sector If θ is an angle in standard position, the region bounded by the initial side and the terminal side of θ, together with the intercepted arc, is called a sector of a circle.
55 Did you know? The area of the sector is proportional to the area of the circle of the same radius. If A is the area of this sector, and θ is the measure of the corresponding central angle in radians, then the ratio of the area of the sector and the area of the circle is proportional to the ratio of θ to 2π radians, that is,
56 Formula for Finding the Area of a Sector A 1 r 2 2
57 Example A central angle of measure 4 intercepts an arc of a circle whose radius is 8cm. Find the length of the intercepted arc.
58 Final Answer: The length of the intercepted arc is 2 centimeter s.
59 Example Two points on the surface of the earth are 5,530 miles apart. If the radius of the earth is approximately 3,950 miles, find the measure of the central angle intercepted by the arc joining the two points on the earth surface.
60 Final Answer: The measure of the central angle intercepted by the arc joining the two points on earth surface is 1.4 radians.
61 Example Find the area of a sector of a circle of radius 4 cm which subtends an angle of 2 radians 3.
62 Final Answer: The area of a sector of a circle is 16 cm 2 3.
63 Something to think about How do we find the relationship between the degree and the radian measures of an angle?
64 The Relationship To find the relationship between the degree and the radian measures of an angle, observe that when the terminal ray makes one complete revolution, the measure of the angle in degrees is 360, while the length of the intercepted arc is equal to the circumference of the circle, which has the value 2πr.
65 Conversions between Degrees and Radians: To convert degrees to radians, multiply the given measure by: 180
66 Conversions between Degrees and Radians: To convert radians to degrees, multiply the given measure by: 180
67 Example 8.2.7: Convert from radians to degrees: 6
68 Final Answer: The radians to degrees is: 30
69 Example 8.2.8: Convert from radians to degrees: 4 9
70 Final Answer: The radians to degrees is: 80
71 Example 8.2.9: Convert from radians to degrees: 25 12
72 Final Answer: The radians to degrees is: 375
73 Example : Convert from degrees to radians: 60
74 Final Answer: The degrees to radians is: 3
75 Example : Convert from degrees to radians: 210
76 Final Answer: The degrees to radians is: 7 6
77 Example : Convert from degrees to radians: 480
78 Final Answer: The degrees to radians is: 8 3
79 To sum it up If the measure of an angle is given without units, it is understood to be given in terms of radians.
80 Equivalence Between Degrees and Radian Measures of Some Special Angles
81 Classroom Task 8.1: Please answer "Let's Practice (LP)" Number 32.
82 Lecture 8.3: Standard Angle and Coterminal Angle SSMTH1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza
83 Recall: An angle is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the x-axis.
84 Did you know? If the terminal side of an angle in standard position is allowed to continue rotating after it has made one complete revolution, it is possible to obtain angles with measures that are greater than 360 or 2π radians, as well as angles with measures that are less than -360 or -2π radians.
85 Coterminal Angles Coterminal angles are angles that have the same initial side and the same terminal side.
86 Coterminal Angles
87 Something to think about From the previous figure, how many angles are coterminal? Give each of their measures.
88 Something to think about How can we determine an angle that is coterminal to a given angle θ?
89 Answer: We can find an angle that is coterminal to a given angle θ by adding or subtracting 2π or one revolution (or 360 ).
90 Example 8.3.1: Find the coterminal angles and sketch the graph: 7 3
91 Final Answer: The coterminal angle is: 3
92 Example 8.3.2: Find the coterminal angles and sketch the graph: 5 4
93 Final Answer: The coterminal angle is: 3 4
94 Example 8.3.3: Find the coterminal angles and sketch the graph: 7 6
95 Final Answer: The coterminal angle is: 5 6
96 Example 8.3.4: Find the coterminal angles and sketch the graph: 85
97 Final Answer: The coterminal angle is: 275
98 Example 8.3.5: Find the value of the acute angle which is coterminal with: 780
99 Relationship Between the Measures of Two Coterminal Angles in Standard Position In general, two angles in standard position are coterminal if their measures differ by an integral multiple of 360 or 2π radians.
100 Final Answer: The given angle is coterminal with the angle that measure 60
101 Reference Angle Let θ be an angle in standard position, and let OP be the terminal side of this angle. The measure of the acute angle which the terminal side makes with the x-axis is called the reference angle of θ.
102 Reference Angle Reference angle of an angle is the smallest positive angle formed between the terminal side and the x-axis.
103 Something to think about Which among the following angles on the board are reference angle of the given θ?
104 Reference Angle
105 Something to think about How can we find the reference angle to a given angle θ?
106 Answer: We can find the reference angle to a given angle θ by adding or subtracting π or 180.
107 How to Compute the Measure of the Reference Angle of an Angle in Standard Position Terminal Side In Quadrant I Quadrant II Quadrant III Quadrant IV Reference Angle (θ in degrees) θ Reference Angle (θ in radians) θ 2
108 Take Note: If this is not the case, we first subtract the largest integral multiple of 2π or 360 from the measure of θ. If the measure of θ is negative, replace it with a coterminal angle with a positive measure.
109 Take Note: When the reference angle is negative take the absolute value, since a reference angle is always positive.
110 Example 8.3.6: Find the reference angle and sketch the graph: 85
111 Final Answer: Since 85 is an acute angle, thus the reference angle is: 85
112 Example 8.3.7: Find the reference angle and sketch the graph: 1056
113 Final Answer: Since a reference angle is always positive. Thus, the reference angle is: 24
114 Example 8.3.8: Find the reference angle and sketch the graph: 14 3
115 Final Answer: The reference angle is: 3
116 Example 8.3.9: Find the reference angle and sketch the graph: 405
117 Final Answer: Since a reference angle is always positive. Thus, the 45 reference angle is:
118 Example : Find the reference angle and sketch the graph: 17 4
119 Final Answer: The reference angle is 4 radians
120 Classroom Task 8.2: Please answer "Let's Practice (LP) Number 33.
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