Questions: Select all properties that apply to the trigonometric function. f(t)=sin (t) A. The function is odd. B. The domain is all real numbers excluding odd multiples of π/2. C. The period is π. D. The domain is all real numbers. E. The period is 2 π. F. The function is even. G. The domain is all real numbers excluding multiples of π.

Select all properties that apply to the trigonometric function.
f(t)=sin (t)
A. The function is odd.
B. The domain is all real numbers excluding odd multiples of π/2.
C. The period is π.
D. The domain is all real numbers.
E. The period is 2 π.
F. The function is even.
G. The domain is all real numbers excluding multiples of π.

Transcript text: Select all properties that apply to the trigonometric function. \[ f(t)=\sin (t) \] A. The function is odd. B. The domain is all real numbers excluding odd multiples of $\frac{\pi}{2}$. C. The period is $\pi$. D. The domain is all real numbers. E. The period is $2 \pi$. F. The function is even. G. The domain is all real numbers excluding multiples of $\pi$.

Solution

Solution Steps

To determine the properties of the trigonometric function $ f(t) = \sin(t) $, we need to analyze its characteristics:

Odd or Even: A function $ f(t) $ is odd if $ f(-t) = -f(t) $ and even if $ f(-t) = f(t) $. The sine function is known to be odd.
Domain: The sine function is defined for all real numbers, so its domain is all real numbers.
Period: The sine function has a period of $ 2\pi $, meaning it repeats its values every $ 2\pi $.

Step 1: Determine if the Function is Odd or Even

The function $ f(t) = \sin(t) $ is odd because it satisfies the property $ f(-t) = -f(t) $. Specifically, $\sin(-t) = -\sin(t)$.

Step 2: Determine the Domain of the Function

The domain of the sine function is all real numbers, $ \mathbb{R} $. There are no restrictions on the input values for $ \sin(t) $.

Step 3: Determine the Period of the Function

The period of the sine function is $ 2\pi $. This means that the function repeats its values every $ 2\pi $ units.