Problem 9 - Algebra 2

Question

PHYSICAL SCIENCE When a person moves along a circular path, the body leans away from a vertical position. The nonnegative acute angle that the body makes with the vertical is called the angle of inclination and is represented by the equation tan \(\theta=\frac{v^{2}}{g R^{\prime}}\) where \(R\) is the radius of the circular path, \(v\) is the speed of the person in meters per second, and \(g\) is the acceleration due to gravity, 9.8 meters per second squared. Write an equivalent expression using \(\sin \theta\) and \(\cos \theta .\)

Step-by-Step Solution

Answer

\( \sin\theta = \frac{v^2}{g R} \cos\theta \)

Step 1: Recall Trigonometric Identity

Recall the trigonometric identity that relates tangent, sine, and cosine: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \).

Step 2: Substitute into Given Equation

Given the equation \( \tan\theta = \frac{v^2}{g R} \), substitute the identity from Step 1 to get \( \frac{\sin\theta}{\cos\theta} = \frac{v^2}{g R} \).

Step 3: Cross-Multiply to Clear Fraction

Cross-multiply the equation from Step 2: \( \sin\theta \cdot g R = \cos\theta \cdot v^2 \).

Step 4: Rearrange for \( \sin\theta \)

Rearrange the equation to express \( \sin\theta \) in terms of \( \cos\theta \): \( \sin\theta = \frac{v^2}{g R} \cos\theta \).

Step 5: Final Formulation

The equivalent expression using \( \sin\theta \) and \( \cos\theta \) is: \( \sin\theta = \frac{v^2}{g R} \cos\theta \).

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the primary trigonometric functions. It is defined as the ratio of the sine function to the cosine function:\[\tan\theta = \frac{\sin\theta}{\cos\theta}\]This relationship highlights how tangent is linked to the other trigonometric functions, sine and cosine. In terms of right-angled triangles, the tangent of an angle represents the ratio of the side opposite the angle to the side adjacent to it.
In physics, the tangent function is useful for describing angles of inclination and analyzing forces acting at angles. It is helpful in scenarios involving circular motion, as seen in the exercise where the angle of inclination \( \theta \) of a moving person is modeled using tangent. Understanding tangent and its relationships with sine and cosine is critical for solving physics problems that involve angular motion.

Sine Function

The sine function, \( \sin \theta \), is a fundamental trigonometric function that represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Its values range between -1 and 1, representing the vertical component of an angle's role in its triangular setting.
In our exercise, the sine function plays a role when we express \( \tan\theta \) in terms of sine and cosine. In activities involving circular motion, as encountered in physics, sine helps quantify component forces perpendicular to the radial direction. This aspect of sine becomes significant when interpreting circular paths or inclinations.
Overall, mastering the sine function equips students to break down complex trigonometric expressions and apply them to physics scenarios effectively.

Cosine Function

The cosine function, \( \cos \theta \), is the counterpart to the sine function. It is defined as the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. Similar to sine, cosine values range from -1 to 1 and represent the horizontal component of an angle's triangle interpretation.
Understanding the cosine function is essential when working with angular measurements, particularly because it frequently appears alongside the sine function in trigonometric identities like the one in the exercise: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). In physics, cosine describes how much of a force is aligned with a direction, which is crucial in assessing inclined objects or analyzing motion along circular paths.
Developing a strong grasp of cosine aids learners in resolving problems involving angles, demonstrating its importance not only in math but also in practical physics applications.

Physics Applications

Trigonometric identities, like those involving the tangent, sine, and cosine functions, play a significant role in physics applications.

• Circular Motion: Trigonometric concepts help describe the physics of circular motion. For instance, the angle of inclination in circular paths can be expressed through tangent and its related functions. These expressions support analyzing forces like gravity and effects at angles.
• Incline Planes and Forces: Sine and cosine functions help break down forces acting on inclined planes, facilitating calculations of components along or perpendicular to the surface.
• Harmonic Motion: The sinusoidal nature of sine and cosine models periodic motion, crucial in understanding oscillations and waves.

Learning to apply these functions helps convert mathematical theories into practical solutions in physics contexts. These concepts form the basis for predicting and interpreting physical phenomena, opening doors to deeper scientific investigations.