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## Chapter 6

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**Chapter 6**Trigonometric Functions**Section 6.1**Angles and Their Measurement**Definition**• An angle is said to be in standard position if its vertex lies at the origin of a cartesian coordinate system and the initial side lies on the positive x-axis. Terminal Side y x Vertex Initial Side**Definition**• Let be an angle in standard position obtained by a counterclockwise rotation. The angle is said to have a measure of one radian if the subtended arc is equal to the radius in length.**Remarks**• The radian measure of an angle is dimensionless. • Since the circumference of a circle of radius r is 2r, an angle that covers an entire circle has measure 2 radians. • For simplicity, we shall often use the unit circle when considering (radian) measures of angles. • If the angle is obtained by a clockwise rotation, then we take the corresponding negative value as the radian measure.**Definition**• One degree (1o) is defined as the measure of the angle subtended by an arc of a circle which is 1/360 of the circumference. • Also, one minute (written 1’) is 1/60 of 1o and one second (written 1’’) is 1/60 of 1’.**Remarks**• If the measure of an angle does not have the degree symbol explicitly indicated, it is assumed to be in radians. • From the definition, we see that the degree measure of an angle which covers an entire circle is 360o. Hence 1 radian = (180/)o 1 degree = (/180) radians**Example 1**• Convert the ff. degree measurements to the equivalent radian measurements: (a) 445o (b) -70o (c) 45o16’48’’**Example 2**• Convert the ff. radian measurements to the equivalent degree measurements: (a) (b) -3**Definition**• An angle in standard position whose terminal side lies on either the x-axis or the y-axis is called quadrantal. • Two angles are coterminal if they have the same terminal side.**Example**• Find the radian measure of the smallest positive angle that is coterminal to the ff. angle having the given radian measure. (a) (b)**Section 6.2**Trigonometric Functions of General Angles**Definition**• Let be an angle in standard position, P(x,y) any point on the terminal side of other than the origin, and the distance from P to the origin. Then,**Remark**• From the definitions, • sin and csc are > 0 only in quadrants I and II • cos and sec are > 0 only in quadrants I and IV • tan and cot are > 0 only in quadrants I and III**Example 1**• Find the values of the trigonometric functions of in standard position if P(12,-6) is on the terminal side of .**Example 2**• Given the ff. information, find the exact values of the remaining trigonometric functions of : (a) csc = 4 and tan < 0 (b) cos = -15/17 and sin < 0**Definition**• An equation that is always true whenever the expression on both sides are defined is called an identity.**Sections 6.3, 6.4**Trigonometric Functions of Special Angles and Real Numbers**Example 1**• Find the exact values of the six trigonometric functions for each angle given below: (a) (d) -90o (b) (e) 3 (c) 480o**Example 2**• Give the exact values of the following without using a calculator: (a) (b)**Theorem***• For any real number t, sin(-t) = -sin t cos(-t) = cos t tan(-t) = -tan t (if tan t is defined)**Theorem***• If tR and kZ, then sin(t + 2k) = sin t cos(t + 2k) = cos t tan(t + k) = tan t (if tan t is defined)**Example**• If f(x) = csc x and f(a) = 2, find the exact value of the following: (a) f(-a) (b) f(a) + f(a + 2) + f(a + 4)**Remarks**• The previous theorem shows that the trigonometric functions are periodic. • A function is said to be periodic if there exists a positive real number p such that for any x in the domain of f, f(x + p) = f(x). The smallest such p is called the period of f. • We can see that sin, cos, sec, and csc each have period 2, while tan and cot have period .**Exercise (no calculators!)**• Convert the ff. angles to the corresponding radian or degree measure (whichever is applicable). Then, find the values of the six trigonometric functions for each angle. 1. 17/2 4. -17/4 2. 635/3 5. 900o 3. -210o**Sections 6.5, 6.6**Graphs of the Trigonometric Functions**Definition**• The graph of a periodic function on an interval of length equal to the period is called a complete cycle, or simply a cycle, of the graph. • In this section, we shall sketch the graphs of the trigonometric functions on (at least) two cycles of their graphs.**Sine Function: Properties**• Its domain is the set of all real numbers. • Its range is the set [-1,1]. • It is periodic with period 2. Hence, once we have drawn the graph for any interval of length 2, say from - to , then this portion will be repeated in intervals of length 2 on the x-axis. • The graph of the sine function is symmetric with respect to the origin.**Cosine Function: Properties**• Its domain is the set of all real numbers. • Its range is the set [-1,1]. • It is periodic with period 2. • The graph of the cosine function is symmetric with respect to the y-axis.**Sine/Cosine Applets**• http://www.math.admu.edu.ph/ma18/web_act/GeogebraActivities-Jume/graphofsine.html • http://www.math.admu.edu.ph/ma18/web_act/GeogebraActivities-Jume/graphofcosine.html**Amplitude of a Wave**• The amplitude of a wave is the maximum displacement of a periodic wave. • If a function is of the form or , then the graph has amplitude |a|.**Examples**• Find the amplitudes of the following functions and sketch their graphs. (a) f(x) = -2cos x (b) g(x) = 1/3 sin x**Period of a Wave**• If or , then the graph has period 2/|b|. • If b < 0, we can use the identities sin(-t) = -sin t and cos(-t) = cos t to simplify the expression first.**Examples**• Find the period and amplitude of the following functions and then sketch their graphs: (a) y = sin(2x) (b) y = -1/2 cos(3x)**Phase Shift**• If y = sin b(x – c), then its graph can be obtained from the graph of y = sin bx by a horizontal translation of |c| units to the right if c > 0 and to the left if c < 0. • The number c is called the phase shift of the graph. • A similar method can be used to obtain the graph of y = cos b(x – c) from y = cos bx.**Examples**• Find the phase shift, amplitude, and period of the ff. functions, and then sketch their graphs. (a) (b) (c)**Exercise**• Work on the exercises in the following links: • http://www.math.admu.edu.ph/ma18/web_act/GeogebraActivities-Jume/Exercise_1___Sine.html • http://www.math.admu.edu.ph/ma18/web_act/GeogebraActivities-Jume/Exercise_1___Cosine.html**Tangent Function: Properties**• Its domain is the set {x | cos x ≠ 0}, or . • It is periodic with period and is symmetric with respect to the origin. • The line x = k/2 (k an odd integer) is a vertical asymptote of the graph. • As x (k/2)+, tan x -. • As x (k/2)-, tan x +.**Cotangent Function: Properties**• Its domain is the set {x | sin x ≠ 0}, or . • It is periodic with period and is symmetric with respect to the origin. • The line x = k (k an integer) is a vertical asymptote of the graph. • As x (k)+, cot x +. • As x (k)-, cot x -.**Remarks**• For a function of the form or , we can sketch its graph using a similar method as that of the corresponding sine and cosine functions. • Note that, in this case, the period will be /|b| and the phase shift is |c|.**Examples**• Sketch the graphs of the following functions over two periods. Indicate the period, phase shift, and domain for each. (a) (b)