Question

Compare \(y=\sin x\) and \(y=\sqrt{1-\cos ^{2} x}\) by completing the following. a) Verify that \(\sin x=\sqrt{1-\cos ^{2} x}\) for \(x=\frac{\pi}{3}, x=\frac{5 \pi}{6},\) and \(x=\pi\). b) Graph \(y=\sin x\) and \(y=\sqrt{1-\cos ^{2} x}\) in the same window. c) Determine whether \(\sin x=\sqrt{1-\cos ^{2} x}\) is an identity. Explain your answer.

Short answer

a) Verified for all given values.b) Both graphs overlap perfectly.c) Not an identity as equality does not hold for all x.

Step-by-step solution

Verify for x = \( \frac{\pi}{3} \)

First, calculate \( \sin \left( \frac{\pi}{3} \right) \). We know from trigonometric values that \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \). Now, calculate \( \sqrt{1 - \cos^2 \left( \frac{\pi}{3} \right)} \). We know \( \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \), so \( 1 - \cos^2 \left( \frac{\pi}{3} \right) = 1 - \left( \frac{1}{2} \right)^2 = 1 - \frac{1}{4} = \frac{3}{4} \). Thus, \( \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \). Therefore, \( \sin \left( \frac{\pi}{3} \right) = \sqrt{1 - \cos^2 \left( \frac{\pi}{3} \right)} \).

Verify for x = \( \frac{5 \pi}{6} \)

First, calculate \( \sin \left( \frac{5 \pi}{6} \right) \). We know \( \sin \left( \frac{5 \pi}{6} \right) = \frac{1}{2} \). Now, calculate \( \sqrt{1 - \cos^2 \left( \frac{5 \pi}{6} \right)} \). We know \( \cos \left( \frac{5 \pi}{6} \right) = -\frac{\sqrt{3}}{2} \), so \( 1 - \cos^2 \left( \frac{5 \pi}{6} \right) = 1 - \left( -\frac{\sqrt{3}}{2} \right)^2 = 1 - \frac{3}{4} = \frac{1}{4} \). Thus, \( \sqrt{\frac{1}{4}} = \frac{1}{2} \). Therefore, \( \sin \left( \frac{5 \pi}{6} \right) = \sqrt{1 - \cos^2 \left( \frac{5 \pi}{6} \right)} \).

Verify for x = \( \pi \)

First, calculate \( \sin (\pi) \). We know \( \sin (\pi) = 0 \). Now, calculate \( \sqrt{1 - \cos^2 (\pi)} \). We know \( \cos (\pi) = -1 \), so \( 1 - \cos^2 (\pi) = 1 - (-1)^2 = 1 - 1 = 0 \). Thus, \( \sqrt{0} = 0 \). Therefore, \( \sin (\pi) = \sqrt{1 - \cos^2 (\pi)} \).

Graphing

Graph both \( y = \sin x \) and \( y = \sqrt{1 - \cos^2 x} \) on the same coordinate plane. Use graphing software or a calculator to visualize the curves. Notice that both graphs overlap perfectly within the range you can observe, indicating they are visually the same.

Determine Identity

To determine whether \( \sin x = \sqrt{1 - \cos^2 x} \) is an identity, consider that for the principle value of \( \sqrt{1 - \cos^2 x} \), it is always non-negative. However, \( \sin x \) can be negative based on the angle \( x \). Thus, \( \sin x = \sqrt{1 - \cos^2 x} \) is not an identity because equality does not hold for all \( x \).

Key concepts

Sine Function

The sine function, denoted as \( \sin x \), is one of the primary functions in trigonometry. It's a periodic function, which means it repeats its values in regular intervals. The sine function is defined for all real numbers and it maps any angle x to a value between -1 and 1. This value is the y-coordinate of a point on the unit circle at that angle. The general form is given by:\(\y = \sin x\). Some key points to remember:

• \sin 0 = 0\
• \sin \frac{\pi}{2} = 1\
• \sin \pi = 0\
• \sin \frac{3\pi}{2} = -1\
• \sin 2\pi = 0\

The function is continuous and smooth, with no holes or jumps, making it easy to graph.

Cosine Function

The cosine function, denoted as \( \cos x \), is another fundamental trigonometric function. It is closely related to the sine function and also periodic, with a range from -1 to 1. The cosine of an angle is the x-coordinate of a point on the unit circle. The general form is given by: \(\y = \cos x\). Important points to remember:

• \cos 0 = 1\
• \cos \frac{\pi}{2} = 0\
• \cos \pi = -1\
• \cos \frac{3\pi}{2} = 0\
• \cos 2\pi = 1\

The cosine function is also continuous and smooth. Unlike the sine function, it starts at its maximum value of 1 when x is 0.

Graphs of Trigonometric Functions

When graphing trigonometric functions like sine and cosine, there are a few important steps and properties to keep in mind. These functions produce waves that can be visually represented on a coordinate plane. The sine and cosine functions both have the following characteristics:

• Periodicity: Both functions repeat their values in intervals of \(2\pi \). This means \( \sin(x + 2\pi) = \sin x \) and \(\cos(x + 2\pi) = \cos x\).
• Amplitude: The maximum and minimum values are 1 and -1, respectively. This means the height of the wave is 2 (from -1 to 1).
• Zeros: The points where the function crosses the x-axis.
  • For \( \sin x \): every \(\pi\).
  • For \( \cos x \): every \( \pi \), starting from zero (0, \Equation: ...).
  • Symmetry: Sine function is an odd function (\sin(-x) = -\sin x). Cosine function is an even function (\cos(-x) = \cos x).
  
  When plotting both \eqref{equation} can be combined for cleaner graph visualization.