Problem 78 - A Graphical Approach to Precalculus with Limits

Question

In the screen shown, the value 3 is stored in S. Then the value of \((\cos (\mathrm{S}))^{2}+(\sin (\mathrm{S}))^{2}\) is shown to be \(1 .\) Duplicate this screen on your own calculator, but use several different values of S. Is the result always \(1 ?\) Explain. (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

Yes, the result is always 1, because \(\cos^2(S) + \sin^2(S) = 1\) is a fundamental trigonometric identity.

Step 1: Understand the Trigonometric Identity

Recognize that the expression \(\cos^2(S) + \sin^2(S)\) represents a fundamental trigonometric identity, which states that for any angle \(S\), the sum of the square of the cosine and sine of that angle is always 1.

Step 2: Choose Your Values

Select different values for \(S\). You can choose any values, such as \(S = 0\), \(S = \frac{\pi}{4}\), \(S = \frac{\pi}{2}\), \(S = \pi\), \(S = 2\pi\), and so on. The choice of \(S\) can be any real number, as this identity applies to all real numbers.

Step 3: Apply the Values to the Identity

For each chosen value of \(S\), substitute \(S\) into the expression \(\cos^2(S) + \sin^2(S)\) and evaluate it using your calculator.For example, if you choose \(S = 0\), then \(\cos^2(0) + \sin^2(0) = 1^2 + 0^2 = 1\).If you choose \(S = \frac{\pi}{4}\), then \(\cos^2\left(\frac{\pi}{4}\right) + \sin^2\left(\frac{\pi}{4}\right) = \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1\).

Step 4: Verify the Results

Observe your calculations for the various values of \(S\). You should find that regardless of what value you choose for \(S\), the result of \(\cos^2(S) + \sin^2(S)\) will always equal 1.

Step 5: Conclusion

Conclude that the result of \(\cos^2(S) + \sin^2(S) = 1\) is always true due to this being a fundamental identity in trigonometry. This holds for any real number \(S\) you choose.

Key Concepts

Cosine Function

A cosine function is a trigonometric function that relates the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. It is denoted as \( \cos(S) \) in trigonometry. Cosine functions are periodic with a period of \( 2\pi \), meaning they repeat their values in a regular cycle every \( 2\pi \) radians.

Key properties of the cosine function include:

• The maximum value of a cosine function is 1, and the minimum is -1.
• Cosine of 0 is 1: \( \cos(0) = 1 \).
• The cosine function is an even function, meaning \( \cos(-S) = \cos(S) \).

This function is crucial in understanding wave patterns, oscillations, and circles. In the equation \( \cos^2(S) + \sin^2(S) = 1 \), the square of the cosine function contributes a part, which when added to the square of the sine function, gives a constant value showcasing the Pythagorean identity in trigonometry.

Sine Function

The sine function is another fundamental trigonometric function, which associates the angle in a right triangle to the ratio of the opposite side over the hypotenuse. It is represented as \( \sin(S) \) and, like the cosine function, is periodic with a period of \( 2\pi \).

Features of the sine function include:

• The range of sine values lies between -1 and 1.
• The sine of 0 is 0: \( \sin(0) = 0 \).
• Sine is an odd function, implying that \( \sin(-S) = -\sin(S) \).

The sine function is pivotal in describing circular motion and waves. In the identity \( \cos^2(S) + \sin^2(S) = 1 \), the sine squared term works with the cosine squared term to demonstrate that regardless of the angle, their sum always equals one.
This identity is rooted in the Pythagorean theorem, as it essential stems from a unit circle definition of sine and cosine.

Real Numbers

Real numbers encompass all the numbers that can be found on the number line, which includes both rational and irrational numbers. They are essential in trigonometry for describing angles and functions, as trigonometric identities like \( \cos^2(S) + \sin^2(S) = 1 \) apply to any real number value of \( S \).

Characteristics of real numbers in trigonometric context:

• Real numbers can be finite decimal expansions (e.g., 3.5, -12) or infinite (e.g., \( \frac{\pi}{4} \)).
• Trigonometric functions like sine and cosine map these real number inputs onto the unit circle.
• The domain for basic trigonometric functions is all real numbers, making them applicable in various real-world and theoretical scenarios.

Understanding that trigonometric identities hold for all real numbers helps ensure the universality of these mathematical truths. You can explore and verify the identity \( \cos^2(S) + \sin^2(S) = 1 \) by choosing diverse real numbers and observing the consistency of the result, showcasing the comprehensive nature of trigonometry.