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I was studying trigonometry when I stumbled upon a problem-- finding out when will a given sinusoidal function reach its first maximum point.

The function is

$$ f(\mathbf t) = 5-2 \sin({2π (\mathbf t+1)\over 7})$$

where t is in seconds, and $\mathbf t \geq 0$

Inputting the function on a graphic calculator shows the function reaches its first maximum point at 4.25 seconds.

Since the coefficient of the function is negative, the first maximum point should correspond to the first minimum point of $\sin x$, which is when $x$ is ${3π \over 2} rad$ or $-{π \over 2} rad.$

Equating: $${2π (\mathbf t+1)\over 7} = {3π \over 2}$$

gives you t = 4.25 seconds, which is correct. Meanwhile, equating: $${2π (\mathbf t+1)\over 7} = {-π \over 2}$$ gives you a negative value, t = -2.75 seconds.

While both corresponds to a highest point, I wonder why they have different values despite ${-π \over 2} rad$ and ${3π \over 2} rad$ essentially being the same angle.

Am I wrong assuming the angles ${-π \over 2} rad$ and ${3π \over 2} rad$ are equivalent?

IceFloat0103
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  • They differ by $2\pi$ and so when plotted in standard position have the same terminal sides. Of course the two numbers in question are different. – coffeemath Jul 20 '22 at 09:20
  • There is a difference between the total amount something has rotated, and its current angle. I'm sure a gymnast doing a double salto will take issue with you saying that it is equivalent to just standing there. As positional angles they are the same, but if interpreted as rotations, they are different. It depends on context. – Jaap Scherphuis Jul 20 '22 at 09:22
  • I see, thank you! When they are treated as rotations, it makes sense. Although they are the same angle, they rotated at different directions hence the difference in the sign. I somehow missed that 3π/2 and 7π/2 are also coinciding, but not necessarily equivalent, especially in this context. – IceFloat0103 Jul 20 '22 at 09:33
  • Angles are equivalent in the meaning, that rotation by both these angles leads to the same result. But if you rotate something with constant angular velocity, you can pass the same position in many moments. These moments are equivalent (in some meaning) but not the same. – Ivan Kaznacheyeu Jul 20 '22 at 09:54
  • Two different numbers measuring the same angle, like saying that 1mt is equal to 100cm (that does not mean that 1 and 100 are the same number). – Mauro ALLEGRANZA Jul 20 '22 at 10:13
  • Use degrees (more familiar), and slowly ponder the full meaning of this sentence: $2^\circ$ and $362^\circ$ are distinct representations of the same angle, in other words, they have different numerical values but are equivalent to each other. – ryang Jul 20 '22 at 10:22

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