The Sine Wave The sine function has domain the set of all real numbers: ( −∞ , ∞ ) but the range is just [ − 1 , 1] since all y -coordinates on the unit circle must be between − 1 and 1. Elementary Functions Part 4, Trigonometry Lecture 4.3a, The Sine Wave Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 1 / 29 Smith (SHSU) Elementary Functions 2013 2 / 29 Cosine Symmetries of sine and cosine Similarly, the domain of cosine is ( −∞ , ∞ ) and the range is [ − 1 , 1] . Let’s consider the definition of sine and cosine on the unit circle and ask about symmetries. Are either of these functions even? odd? We assume that a positive angle θ involves a counterclockwise rotation starting at the point P (1 , 0) and moving above the x -axis, while a negative value of θ means we move clockwise, starting at the point P (1 , 0) and moving below the x -axis, Is it clear that moving clockwise instead of counterclockwise does not change the sign of the x -value of the point P ( x, y ) ? That is, for any angle θ , cos( − θ ) = cos( θ ) and so cosine is an even function. Smith (SHSU) Elementary Functions 2013 3 / 29 Smith (SHSU) Elementary Functions 2013 4 / 29
Symmetries of sine and cosine Central Angles and Arcs Here is the graph of the sine function. Notice the rotational symmetry about the origin. But the sine function has much more symmetry than just rotational symmetry about the origin. It is in fact periodic with period 2 π ( ≈ 6 . 28 . ) However, if we begin to move clockwise around the origin, beginning on Since 2 π radians makes a complete revolution of the circle then the x -axis at (1 , 0) then the y -value of the point P ( x, y ) immediately becomes negative instead of positive. sin( θ + 2 π ) = sin( θ ) . Reversing the direction of rotation reverses the sign of the y -value and so sin( − θ ) = − sin( θ ) . Therefore the sine function is odd. Smith (SHSU) Elementary Functions 2013 5 / 29 Smith (SHSU) Elementary Functions 2013 6 / 29 Some worked problems The Sine Wave Solve each of the following equations. 1 Describe the set of all the angles θ that satisfy the trig equation 1 cos x = 1 √ 3 sin θ = − 2 . 2 2 cos x = 1 3 (2 cos x − 1)(cos x − 1) = 0 . Solution. If the sine of an angle is negative then it must be in the third of fourth quadrants. From our knowledge of 30-60-90 triangles, we see that θ = 4 π 3 ( 240 ◦ ) and θ = 5 π Solutions. 3 ( 300 ◦ ) are angles whose sine 1 Since cos(0) = 1 then cos x = 1 means that x is either 0 or 0 plus √ 3 is − 2 . some multiple of 2 π. We can write this all in the form 0 + 2 πk (where k is an integer) or Since the sine function is periodic these are not the only solutions to { 2 πk : k ∈ Z } . this equation! Since the sine function is periodic with period 2 π we know that 3 = 1 2 Since cos π 2 then x = π 3 is a solution to 2 cos x = 1 . So also is θ = 4 π 3 + 2 πk and θ = 5 π x = − π 3 . (Remember, f ( x ) = cos x is an even function!) Since the 3 + 2 πk , period of cosine is 2 π then our set of all solutions is (where k is an integer) will also be solutions. { π 3 + 2 πk : k ∈ Z } ∪ {− π 3 + 2 πk : k ∈ Z } . Smith (SHSU) Elementary Functions 2013 7 / 29 Smith (SHSU) Elementary Functions 2013 8 / 29
The Sine Wave Amplitude, period and phase shift In practical applications many periodic functions are transformations of the sine function. A transformation of the sine function is often called a sine wave or a sinusoid. In general, sine waves will have form 3 Solve the equation (2 cos x − 1)(cos x − 1) = 0 . f ( θ ) = a sin( b ( θ − c )) + d. (1) Solution. Any solution to (2 cos x − 1)(cos x − 1) = 0 is either a solution to 2 cos x − 1 = 0 or a solution to cos x − 1 = 0 . From our earlier discussion of transformations, we see that one can transform the graph of sin( θ ) into the graph of f ( θ ) = a sin( b ( θ + c )) + d We have already solved these equations in the previous two problems. by the following steps (in this order!): All of the solutions the previous two problems are solutions to this 1 Shift right by c , problem. So our answer is 2 Shrink horizontally by a factor of b (about the line x = c ), { 2 πk : k ∈ Z } ∪ { π 3 + 2 πk : k ∈ Z } ∪ {− π 3 + 2 πk : k ∈ Z } . 3 Expand vertically by a factor of a 4 Shift up by d . Some of these translations are associated with particular terms. We revisit the concept of period and introduce new terms frequency, amplitude and phase shift . Smith (SHSU) Elementary Functions 2013 9 / 29 Smith (SHSU) Elementary Functions 2013 10 / 29 Period of a sine wave Frequency of a sine wave The period of a sine wave tells us how many units of the input variable are required before the function repeats. Since the sine function has period 2 π then the sine wave given by the function f ( θ ) = a sin( b ( θ − c )) + d will have period 2 π The frequency a sine wave is the number of times the wave repeats within | b | . a single unit of the input variable θ ; this is the reciprocal of the period. (We use an absolute value sign here since we want the period to be positive and it is possible that b is negative.) Thus the frequency of the standard sine wave sin( x ) is 1 2 π and so the frequency of f ( θ ) = a sin( b ( θ − c )) + d is | b | 2 π. Smith (SHSU) Elementary Functions 2013 11 / 29 Smith (SHSU) Elementary Functions 2013 12 / 29
Frequency Amplitude of a sine wave Electronic transmissions involve the sine wave. The frequency of the The height of the standard sine wave oscillates between a maximum of 1 transmission represents the number of copies of the sine wave which occur and minimum of − 1 . within a single unit of time (often one second.) If we consider the midpoint of this wave, then the wave rises 1 unit above For example, an electromagnetic wave with frequency 4 . 3 × 10 14 oscillates and then drops 1 unit below this midpoint. 430 , 000 , 000 , 000 , 000 (430 million million) times in one second and is perceived by our eyes as the color red. This variation from the “average” height is the amplitude of the sine wave. For the standard sine wave the amplitude is 1. Scientists often use the term “hertz” to represent “cycles per second” and so we say that frequency of red light is 4 . 3 × 10 14 hertz. The amplitude of the sine wave f ( θ ) = a sin( b ( θ − c )) + d is just | a | . (Again, we use absolute value because we want the amplitude to be A light wave of lower frequency will not be visible to our eyes; waves of positive.) higher frequency will show up as orange, yellow, green, and so on. Smith (SHSU) Elementary Functions 2013 13 / 29 Smith (SHSU) Elementary Functions 2013 14 / 29 Phase shift of a sine wave The sine wave The graph of the standard sine wave sin( θ ) passes through the origin (0 , 0) . In the next presentation, we will continue to look further at sine waves (sinusoids) including the cosine function. A sine wave might be shifted to the right by an amount c ; this is the phase shift of the sine wave f ( θ ) = a sin( b ( θ − c )) + d. (End) Note that the phase shift can be negative. A negative phase shift means that the graph of sin θ is being shifted by a certain amount to the left. Smith (SHSU) Elementary Functions 2013 15 / 29 Smith (SHSU) Elementary Functions 2013 16 / 29
The graph of cosine . The graph of cosine function has a very similar wave pattern to that of sine. Here is a graph of the cosine function. Elementary Functions Part 4, Trigonometry Lecture 4.3b, The Cosine and other Sinusoids Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 17 / 29 Smith (SHSU) Elementary Functions 2013 18 / 29 Central Angles and Arcs The symmetries of the six trig functions Just as we did with sine waves, we may consider graphs of Since the sine function is odd and the cosine function is even then tan( − θ ) = sin( − θ ) cos( − θ ) = − sin( θ ) g ( θ ) = a cos( b ( θ − c )) + d. = − tan( θ ) cos( θ ) There is no significant difference in meaning for the period, frequency, and so the tangent function is odd. Here is a graph of the tangent amplitude or phase shift when discussing the cosine function. function: The function g ( θ ) has period p = 2 π | b | , frequency f = | b | 2 π , amplitude | a | and phase shift c . We tend to concentrate on the sine wave and ignore the cosine function. This is merely because the graph of cosine function is really a shift of the graph of sine! A careful examination of the graphs of these functions demonstrate that the graph of cos( θ ) is the graph of sin( θ ) shifted to the left by π 2 . cos( θ ) = sin( θ + π 2 ) . We could think of the cosine function as a sine wave with phase shift − π 2 . Smith (SHSU) Elementary Functions 2013 19 / 29 Smith (SHSU) Elementary Functions 2013 20 / 29
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