When we study trigonometric functions like sine, cosine, and tangent, it’s important to know how they are different in terms of amplitude, period, and phase shift. Let’s break this down into simpler parts.
Sine Function (y = sin(x)): The amplitude is how high or low the wave goes. For sine, the highest point is 1 and the lowest point is -1. So, it fluctuates between -1 and 1.
Cosine Function (y = cos(x)): Just like sine, the amplitude here is also 1. It also goes between -1 and 1.
Tangent Function (y = tan(x)): The tangent function is different. It doesn’t have a fixed amplitude. It can go as high or as low as any number, so its range is from negative infinity to positive infinity.
Sine Function: The period tells us how long it takes for the wave to repeat its pattern. For sine, this period is (2\pi). This means it completely repeats every (2\pi) radians, which is about 6.28 units.
Cosine Function: Cosine also has a period of (2\pi). So, it repeats its pattern at the same interval as sine.
Tangent Function: The period for tangent is shorter, at (\pi). This means it repeats its pattern twice as fast as sine and cosine, every (\pi) radians or about 3.14 units.
Phase shift refers to moving the wave left or right. For the basic sine and cosine functions without changes:
Sine Function: Normally, the sine function starts at 0, which means no phase shift. If we write it as (y = \sin(b(x - d))), the letter (d) tells us how much we shift it.
Cosine Function: The basic cosine function also starts at 0, like sine. It can be shifted using the equation (y = \cos(b(x - d))).
Tangent Function: The standard tangent function starts at 0 too, but we can also express it as (y = \tan(b(x - d))), where (d) indicates how much we shift it.
To sum it all up:
Amplitude: Both sine and cosine have an amplitude of 1, while tangent doesn’t have one.
Period: Sine and cosine repeat every (2\pi), but tangent repeats every (\pi).
Phase Shift: All three functions usually start with a phase shift of 0, unless we change them with some numbers in their equations.
Knowing these differences helps students understand and draw these functions better in math!
When we study trigonometric functions like sine, cosine, and tangent, it’s important to know how they are different in terms of amplitude, period, and phase shift. Let’s break this down into simpler parts.
Sine Function (y = sin(x)): The amplitude is how high or low the wave goes. For sine, the highest point is 1 and the lowest point is -1. So, it fluctuates between -1 and 1.
Cosine Function (y = cos(x)): Just like sine, the amplitude here is also 1. It also goes between -1 and 1.
Tangent Function (y = tan(x)): The tangent function is different. It doesn’t have a fixed amplitude. It can go as high or as low as any number, so its range is from negative infinity to positive infinity.
Sine Function: The period tells us how long it takes for the wave to repeat its pattern. For sine, this period is (2\pi). This means it completely repeats every (2\pi) radians, which is about 6.28 units.
Cosine Function: Cosine also has a period of (2\pi). So, it repeats its pattern at the same interval as sine.
Tangent Function: The period for tangent is shorter, at (\pi). This means it repeats its pattern twice as fast as sine and cosine, every (\pi) radians or about 3.14 units.
Phase shift refers to moving the wave left or right. For the basic sine and cosine functions without changes:
Sine Function: Normally, the sine function starts at 0, which means no phase shift. If we write it as (y = \sin(b(x - d))), the letter (d) tells us how much we shift it.
Cosine Function: The basic cosine function also starts at 0, like sine. It can be shifted using the equation (y = \cos(b(x - d))).
Tangent Function: The standard tangent function starts at 0 too, but we can also express it as (y = \tan(b(x - d))), where (d) indicates how much we shift it.
To sum it all up:
Amplitude: Both sine and cosine have an amplitude of 1, while tangent doesn’t have one.
Period: Sine and cosine repeat every (2\pi), but tangent repeats every (\pi).
Phase Shift: All three functions usually start with a phase shift of 0, unless we change them with some numbers in their equations.
Knowing these differences helps students understand and draw these functions better in math!