A sine function is a common trigonometric function that describes the relative amplitude and period of a wave. It is typically defined by an amplitude, A, and period, T, where:
Y(x) = A sin(x) = A () where means the sine of x.
The term ‘sine’ comes from the English word ‘sinew’ which refers to the thin cord in your arm. Imagine swinging a tennis racket to hit a ball, the amplitude of your hand moving down relative to your forearm is like the amplitude of the wave moving down relative to the length of the tennis racket.
This article will answer questions like: What is Y(4)? For what value of 4 is Y equal to?\r
Sine Function with Amplitude 0.75 and Period 10\r
In this example we will find out what Y(4) is for a specific value of 4. We will do this by finding out what A sin(4) equals and then multiplying that result by 0.75 to get Y(4).
A sin(4) can be calculated using cos(4)-1 by taking the difference between cosine 4 and 1 and then multiplying by -1 since sin(-4) = 1.
A sin(-4) = cos(-4)-1
(-1)(cos(-4)) = -1 cos (- 4 )
(cos (- 4 ) – 1 ) = -1 * 1
(-1)(0.75) = 0.75
(0 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 127 129 131 133 135 137 139 141 143 145 147 149 151 153 155 157 159 161 163 165 167 169 171 173 175 177 179 181 183 185 187 189 191 193 195 197 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243 245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279 281 283 285 287 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 403 405 407 409 411 413 415 417 419 421 423 425 427 429 431 433 435 437 439 441 443 445\r
Rearranging this gives us A sin(-4)=0 . 75 * 1 which simplifies to 0 . 75.
>For any value X, if we know its corresponding angle in degrees (or radians), then we can find out what Y equals! Simply multiply X by A and you have Y.So now that we know that for wave with an amplitude of 0 . 75 , Y ( 4 )= 0 . 75 , or equivalently , for an angle of magnitude 4 , y (that is , shoreline waves wash ashore at average speed of 0 . 75 per second.
The period of this sine function is 10
The period of a function is the length of the variable in which the value changes. In other words, the period is how many units the value changes per unit unit time.
In this case, the period is 10 units, which means that for every ten seconds, the output value changes by one unit. For example, if the function outputs a value of one at time zero, then after ten seconds it will output a value of one again.
The amplitude of this sine function is 0.75
Amplitude refers to the height of the curve relative to the base number. In this case, the amplitude is how tall half of the circle is- 0.75.
This means the amplitude is also 10
The sine function is one of the most important functions in mathematics. It is defined as the ratio of the opposite side of a triangle to its hypotenuse.
This definition can be applied to linear functions as well. A linear function can be defined as the output value relative to the input value. The output value is dependent on the input value, or the length of the line.
The sine function is defined as y = sin(x), where x is the angle passed into the function and y is what comes out. The variable x can be any number, even numbers only!
The sine function has a special property: it oscillates between -1 and 1. This means that when you input an angle into the function, it will come out as either -1 or 1, depending on whether you input an angle pointing clockwise or counterclockwise.
Now we can use the sine function formula to find Y(4)
Y(4) = 0.75sin(4)
Since the period of the sine function is 4, we can say that Y(4) = 0.75sin(0), or Y(4)=0.75 when x=0.
Since the amplitude of the sine function is 0.75, we can say that Y(4)=0.75*0.75 or Y(4)=0.5625 When x=0, or y=−0.5625 when y=−z .
You can try plugging in different values for x to see this for yourself!
Sine functions have a starting and ending point at zero; this is why y equals zero at x equals zero in the last bullet point.
Y(4) = A*sin(10*4) + Y(3)
Now that you understand Y(3), let’s look at how to find the value of Y(4) when the period is 4.
The amplitude of the sine function is 0.75, and the period is 10, so you can use the formula A*sin(10*4) + Y(3).
This formula means that you multiply 0.75 by the sin of 10*4, which is -0.75, and then add the previous value of Y. In this case, Y=0, so you just add 0 to get a final answer of 0.75.
You can check this answer by computing what Y would be at time 4, and seeing if it matches 0.
Y(4) = 0.75*sin(40°) + 0.75*0.7364 + 0.7289
In this article, you learned how to find the value of y(4) for a sine function with amplitude A=0.75 and period T=10. You did this by first finding the amplitude of the fourth y-value, 0.7289, then adding 0.75*0.7364 to that value.
You then found the sine of 40° and multiplied that by 0.75 to get 0.75*sin(40°) before adding both values together. This gave you Y(4) = 0.75*0.7364 + 0.7289, which is the answer!
You can use this knowledge in many ways, such as finding what the fourth y-value is for any given x-value, finding what x corresponds to a given y-value, or finding what x and y are relative to each other.
Y(4)= 0.7431 + 0.3025
In this bullet point, you will learn how to write the formula for y(4) for a sine function with amplitude A=0.75 and period T=10. You will be given the input of 4, which is the number of periods ahead of the starting point at 0.
To start, you must recall that the amplitude is how far up or down the curve goes from 0. The amplitude in this case is how high or low the sine curve goes from 0 to 4, or 0 to whatever other input you have.
Then, you must remember that there are two intervals per period. One goes up, and one goes down. You must also remember that there are two possible outputs per interval: +1 or -1.
Combining these memories with what was given in the question, you can write the formula for y(4) as: y(4)=0.7431+(0.75)(0)+(-0.75)(1)+(-1)(-1)+(-2)(-2)+(-3)(-3)+(-4)(-4)=0+0+0+0+1+1+2+3=10! That’s a lot of zeroes!
Thank you for reading this article! Hope you enjoyed it.
NemesisFazbear82 is a fanfiction author who has written 6 stories for Animorphs, Over The Garden Wall, and The Demon Tower.
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