Sin & Cos with Amplitude and Phase. Trigonometric equations are applied in the world of TV / cable broadcast. Equations such as y=2 sin 𝑥 is an example of such an equation.

Sin & Cos with Amplitude and Phase. Trigonometric equations are applied in the world of TV / cable broadcast. Equations such as y=2 sin 𝑥 is an example of such an equation. In the equation, 2 is a multiplier and called an amplitude. Amplitude describes the “height” of the trigonometric function.

Sin & Cos with Amplitude and Phase. Trigonometric equations are applied in the world of TV / cable broadcast. Equations such as y=2 sin 𝑥 is an example of such an equation. In the equation, 2 is a multiplier and called an amplitude. Amplitude describes the “height” of the trigonometric function. 𝑦=𝐴 𝑠𝑖𝑛 𝑥 Let’s compare to y=2 sin 𝑥 on the interval [ 0 , 2π ] 𝑦=𝑠𝑖𝑛 𝑥

Sin & Cos with Amplitude and Phase. Trigonometric equations are applied in the world of TV / cable broadcast. Equations such as y=2 sin 𝑥 is an example of such an equation. In the equation, 2 is a multiplier and called an amplitude. Amplitude describes the “height” of the trigonometric function. 𝑦=𝐴 𝑠𝑖𝑛 𝑥 Let’s compare to y=2 sin 𝑥 on the interval [ 0 , 2π ] 𝑦=𝑠𝑖𝑛 𝑥 𝑦=𝑠𝑖𝑛 𝑥 2 sin 0𝜋= sin 𝜋 4 = 1 sin 𝜋 2 = sin 3𝜋 4 = sin 𝜋= 𝜋 4 𝜋 2 3𝜋 4 𝜋 5𝜋 4 3𝜋 2 7𝜋 4 2𝜋 sin 5𝜋 4 = -1 sin 3𝜋 2 = sin 7𝜋 4 = -2 sin 2𝜋=

Sin & Cos with Amplitude and Phase. 𝑦=𝐴 𝑠𝑖𝑛 𝑥 I used just basic angles and plotted my sin x curve. Let’s compare to y=2 sin 𝑥 on the interval [ 0 , 2π ] 𝑦=𝑠𝑖𝑛 𝑥 𝑦=𝑠𝑖𝑛 𝑥 2 sin 0𝜋= 0 sin 𝜋 4 =0.7071 1 sin 𝜋 2 = 1 5𝜋 4 7𝜋 4 sin 3𝜋 4 =0.7071 sin 𝜋=0 𝜋 4 𝜋 2 3𝜋 4 𝜋 3𝜋 2 2𝜋 sin 5𝜋 4 =−0.7071 -1 sin 3𝜋 2 =−1 sin 7𝜋 4 =−0.7071 -2 sin 2𝜋=0

Sin & Cos with Amplitude and Phase. 𝑦=𝐴 𝑠𝑖𝑛 𝑥 I used just basic angles and plotted my sin x curve. Now let’s get our values for y=2 sin 𝑥 Let’s compare to y=2 sin 𝑥 on the interval [ 0 , 2π ] 𝑦=𝑠𝑖𝑛 𝑥 𝑦=𝑠𝑖𝑛 𝑥 y=2 sin 𝑥 2 sin 0𝜋= 0 2 sin 0𝜋= sin 𝜋 4 =0.7071 2 sin 𝜋 4 = 1 sin 𝜋 2 = 1 2 sin 𝜋 2 = 5𝜋 4 7𝜋 4 sin 3𝜋 4 =0.7071 2 sin 3𝜋 4 = sin 𝜋=0 2 sin 𝜋= 𝜋 4 𝜋 2 3𝜋 4 𝜋 3𝜋 2 2𝜋 sin 5𝜋 4 =−0.7071 2 sin 5𝜋 4 = -1 sin 3𝜋 2 =−1 2 sin 3𝜋 2 = sin 7𝜋 4 =−0.7071 2 sin 7𝜋 4 = -2 sin 2𝜋=0 2 sin 2𝜋=

Sin & Cos with Amplitude and Phase. 𝑦=𝐴 𝑠𝑖𝑛 𝑥 I used just basic angles and plotted my sin x curve. Now let’s get our values for y=2 sin 𝑥 As you can see, all the values doubled ( x 2 ) Let’s compare to y=2 sin 𝑥 on the interval [ 0 , 2π ] 𝑦=𝑠𝑖𝑛 𝑥 𝑦=𝑠𝑖𝑛 𝑥 y=2 sin 𝑥 2 sin 0𝜋= 0 2 sin 0𝜋=0 sin 𝜋 4 =0.7071 2 sin 𝜋 4 =1.414 1 sin 𝜋 2 = 1 2 sin 𝜋 2 =2 5𝜋 4 7𝜋 4 sin 3𝜋 4 =0.7071 2 sin 3𝜋 4 =1.414 sin 𝜋=0 2 sin 𝜋= 0 𝜋 4 𝜋 2 3𝜋 4 𝜋 3𝜋 2 2𝜋 sin 5𝜋 4 =−0.7071 2 sin 5𝜋 4 =−1.414 -1 sin 3𝜋 2 =−1 2 sin 3𝜋 2 =−1 sin 7𝜋 4 =−0.7071 2 sin 7𝜋 4 = −1.414 -2 sin 2𝜋=0 2 sin 2𝜋= 0

Sin & Cos with Amplitude and Phase. Phase relation is seen in practical applications such as sound, electrical, and radio waves. This “phase shift” adjusts the wave by sliding it either left or right a number of degrees. The waves mostly frequently shifted are sine waves. 𝑦=𝑠𝑖𝑛 𝑥±∅

Sin & Cos with Amplitude and Phase. Phase relation is seen in practical applications such as sound, electrical, and radio waves. This “phase shift” adjusts the wave by sliding it either left or right a number of degrees. The waves mostly frequently shifted are sine waves. 𝑦=𝑠𝑖𝑛 𝑥±∅ Here is an example of a sine wave shifted 45⁰.

Sin & Cos with Amplitude and Phase. Phase relation is seen in practical applications such as sound, electrical, and radio waves. This “phase shift” adjusts the wave by sliding it either left or right a number of degrees. The waves mostly frequently shifted are sine waves. 𝑦=𝑠𝑖𝑛 𝑥±∅ Here is an example of a sine wave shifted 45⁰. ( the interval is [ 0 , 2π ] 𝑦=𝑠𝑖𝑛 𝑥 sin 0𝜋= 0 1 sin 𝜋 4 =0.7071 sin 𝜋 2 = 1 5𝜋 4 7𝜋 4 sin 3𝜋 4 =0.7071 sin 𝜋=0 𝜋 4 𝜋 2 3𝜋 4 𝜋 3𝜋 2 2𝜋 sin 5𝜋 4 =−0.7071 -1 sin 3𝜋 2 =−1 sin 7𝜋 4 =−0.7071 sin 2𝜋=0

Sin & Cos with Amplitude and Phase. Phase relation is seen in practical applications such as sound, electrical, and radio waves. This “phase shift” adjusts the wave by sliding it either left or right a number of degrees. The waves mostly frequently shifted are sine waves. 𝑦=𝑠𝑖𝑛 𝑥±∅ Here is an example of a sine wave shifted 45⁰. ( the interval is [ 0 , 2π ] 𝑦=𝑠𝑖𝑛 𝑥 𝑦=𝑠𝑖𝑛 𝑥+45° sin 0𝜋= 0 sin 0+45° =0.7071 1 sin 𝜋 4 =0.7071 sin 𝜋 4 +45° =1 sin 𝜋 2 = 1 5𝜋 4 7𝜋 4 sin 3𝜋 4 =0.7071 sin 𝜋 2 +45° =0.7071 sin 𝜋=0 𝜋 4 𝜋 2 3𝜋 4 𝜋 3𝜋 2 2𝜋 sin 5𝜋 4 =−0.7071 sin 3𝜋 4 +45° =0 -1 sin 3𝜋 2 =−1 sin 7𝜋 4 =−0.7071 ⋮ sin 2𝜋=0 And so on

Sin & Cos with Amplitude and Phase. Phase relation is seen in practical applications such as sound, electrical, and radio waves. This “phase shift” adjusts the wave by sliding it either left or right a number of degrees. The waves mostly frequently shifted are sine waves. As you can see, we either add or subtract the angle. 𝑦=𝑠𝑖𝑛 𝑥±∅ Here is an example of a sine wave shifted 45⁰. ( the interval is [ 0 , 2π ] 𝑦=𝑠𝑖𝑛 𝑥 𝑦=𝑠𝑖𝑛 𝑥+45° sin 0𝜋= 0 sin 0+45° =0.7071 1 sin 𝜋 4 =0.7071 sin 𝜋 4 +45° =1 sin 𝜋 2 = 1 5𝜋 4 7𝜋 4 sin 3𝜋 4 =0.7071 sin 𝜋 2 +45° =0.7071 sin 𝜋=0 𝜋 4 𝜋 2 3𝜋 4 𝜋 3𝜋 2 2𝜋 sin 5𝜋 4 =−0.7071 sin 3𝜋 4 +45° =0 -1 sin 3𝜋 2 =−1 sin 7𝜋 4 =−0.7071 ⋮ sin 2𝜋=0 And so on