Question

Copy and complete the table of properties for \(y=\sin x\) and \(y=\cos x\) for all real numbers. $$\begin{array}{l|l|l|} \text { Property } & y=\sin x & y=\cos x \\ \hline \text { maximum } & & \\ \hline \text { minimum } & & \\ \hline \text { amplitude } & & \\ \hline \text { period } & & \\ \hline \text { domain } & & \\ \hline \text { range } & & \\ \hline y \text { -intercept } \\ \hline x \text { -intercepts } \\ \hline \end{array}$$

Short answer

Both \(y = \sin x\) and \(y = \cos x\) have a max of 1, min of -1, amplitude of 1, and period of \(2\pi\). Domain: all real numbers. Range: \([-1, 1]\). y-intercepts: 0 (sine), 1 (cosine). x-intercepts: \(n\pi\) (sine), \[(2n+1)\pi/2\] (cosine).

Step-by-step solution

Identify the maximum values

For the function \(y = \sin x\), the maximum value is 1 because the sine function reaches its highest point at 1. Similarly, for the function \(y = \cos x\), the maximum value is also 1 since the cosine function also reaches its highest point at 1.

Identify the minimum values

For the function \(y = \sin x\), the minimum value is -1 because the sine function reaches its lowest point at -1. Similarly, for the function \(y = \cos x\), the minimum value is also -1 since the cosine function also reaches its lowest point at -1.

Determine the amplitude

The amplitude of a trigonometric function is the distance from the midline to the peak of the wave. For both \(y = \sin x\) and \(y = \cos x\), the amplitude is 1 because the peak value is 1 and the midline is 0.

Determine the period

The period of a trigonometric function is the distance over which the function repeats itself. For both \(y = \sin x\) and \(y = \cos x\), the period is \(2\pi\) because the function completes one cycle every \(2\pi\) radians.

Identify the domain

The domain of \(y = \sin x\) and \(y = \cos x\) is all real numbers, because these functions are defined for every real number.

Identify the range

The range of \(y = \sin x\) and \(y = \cos x\) is \([-1, 1]\), because these functions only take values between -1 and 1, inclusive.

Determine the y-intercept

For \(y = \sin x\), the y-intercept is 0 because \(\sin(0) = 0\). For \(y = \cos x\), the y-intercept is 1 because \(\cos(0) = 1\).

Determine the x-intercepts

For \(y = \sin x\), the x-intercepts occur at \(x = n\pi\) where \(n\) is an integer because \(\sin(n\pi) = 0\). For \(y = \cos x\), the x-intercepts occur at \(x = (2n+1)\pi/2\) where \(n\) is an integer because \(\cos((2n+1)\pi/2) = 0\).

Key concepts

Sine Function

The sine function, represented as \(y = \sin x\), is one of the fundamental trigonometric functions. It describes a smooth, periodic wave. The full name is sine of x.
The sine function:

• Oscillates between -1 and 1
• Has a period of \(2\pi\)
• Is symmetric about the origin due to its odd nature

The sine curve starts at 0 when \(x = 0\), reaches its maximum of 1 at \(x = \pi/2\), returns to 0 at \(x = \pi\), hits its minimum of -1 at \(x = 3\pi/2\), and completes the cycle at \(2\pi\). Understanding sine helps in solving many real-world problems, especially in physics and engineering, like modeling sound waves and light waves.

Cosine Function

The cosine function, represented as \(y = \cos x\), is another essential trigonometric function. It is similar to the sine function but shifted.
The cosine function:

• Also oscillates between -1 and 1
• Has the same period of \(2\pi\)
• Is symmetric about the y-axis, making it an even function

For the cosine function, the wave begins at 1 when \(x = 0\), drops to 0 at \(x = \pi/2\), reaches -1 at \(x = \pi\), returns to 0 at \(x = 3\pi/2\), and comes back to 1 at \(2\pi\). This shift is half a period ahead of the sine function. Understanding cosine is vital for tasks in engineering and computer science, like signal processing.

Amplitude

Amplitude is a measure of the maximum deviation of a function from its average value. In simpler terms, it indicates how 'tall' the wave is from its center.

• For both \(y = \sin x\) and \(y = \cos x\), the amplitude is 1.
• This is because the maximum value is 1, and the minimum value is -1.
• The distance from the center (which is zero) to the maximum point is always 1.

So, the amplitude is always positive and tells you the height of the peaks of the wave from the middle line. In applications like sound and light, amplitude is related to how loud a sound is or how bright a light is.

Period

The period of a trigonometric function is the length it takes for the function to complete one full cycle.
For both \(y = \sin x\) and \(y = \cos x\):

• The period is \(2\pi\)
• This means every \(2\pi\) units along the x-axis, the function values repeat.

This is crucial for understanding oscillations and waves. A shorter period means the wave oscillates more rapidly, while a longer period means slower oscillations. For example, knowing the period helps in analyzing cyclical phenomena like tides and seasonal changes.

Domain and Range

The domain and range are fundamental concepts for understanding the full behavior of a function.

• The domain of both \(y = \sin x\) and \(y = \cos x\) is all real numbers: \( -\infty < x < \infty \).
• This means you can plug in any real number into the function, and it will return a value.

As for the range:

• The range for both functions is \([-1, 1]\)
• The functions will only output values within this interval, meaning they oscillate between -1 and 1, inclusive.

Understanding the domain and range is essential for graphing these functions and applying them in real-world problems such as engineering and physics. It tells you the complete set of possible inputs and outputs for the function.