Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Back Stress If a person bends at the waist with a straight back, making an angle of \(\theta\) degrees with the horizontal, then the force \(F\) exerted on the back muscles can be modeled by the equation $$F=\frac{0.6 W \sin \left(\theta+90^{\circ}\right)}{\sin 12^{\circ}}$$ where \(W\) is the weight of the person. (Source: Metcalf, H., Topics in Classical Biophysics, Prentice- Hall.) (a) Calculate \(F\) when \(W=170\) pounds and \(\theta=30^{\circ}\) (b) Use an identity to show that \(F\) is approximately equal to \(2.9 \mathrm{W} \cos \theta\) (c) For what value of \(\theta\) is \(F\) maximum?

Step-by-Step Solution

Answer

(a) Calculate using specific values. (b) Prove the approximation with an identity. (c) Max value when \(\theta = 0^{\circ}\).

Step 1: Substitute given values into the equation

For part (a), we are provided with \(W = 170\) pounds and \(\theta = 30^{\circ}\). Substitute these values into the equation for \(F\):\[F = \frac{0.6 \times 170 \times \sin(30^{\circ} + 90^{\circ})}{\sin 12^{\circ}}.\]

Step 2: Simplify using trigonometric identities

Calculate \(\sin(120^{\circ})\) since \(30^{\circ} + 90^{\circ} = 120^{\circ}\). We know that \(\sin(120^{\circ}) = \sin(180^{\circ} - 60^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2}\).

Step 3: Calculate F

Substitute \(\sin(120^{\circ}) = \frac{\sqrt{3}}{2}\) and \(\sin(12^{\circ})\) into the equation:\[F = \frac{0.6 \times 170 \times \frac{\sqrt{3}}{2}}{\sin 12^{\circ}}.\]Use a calculator to find \(F\).

Step 4: Use identity to prove the approximation

For part (b), use the identity \(\sin(\theta + 90^{\circ}) = \cos \theta\) to rewrite the equation as:\[F = \frac{0.6 W \cos \theta}{\sin 12^{\circ}}.\]Given \(\frac{0.6}{\sin 12^{\circ}} \approx 2.9\), we can state: \[F \approx 2.9 W \cos \theta.\]

Step 5: Determine maximum value of F

For part (c), the function has its maximum value when \(\cos \theta\) is maximized. Since -1 to 1 is the range for \(\cos \theta\), \(\cos \theta\) reaches its maximum value of 1 when \(\theta = 0^{\circ}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the variable. They help us simplify expressions and solve trigonometric equations more easily. In this exercise, you can see how a trigonometric identity can transform a problem involving sine into one involving cosine.

One essential identity used here is the relationship

• \(\sin(\theta + 90^\circ) = \cos \theta\)

This identity takes advantage of the special angle relationship, where shifting an angle by 90 degrees converts a sine function to a cosine function.

Understanding trigonometric identities is crucial for breaking down complex trigonometric equations into simpler forms. This allows for easier calculations and insights into the relationships of the trigonometric functions involved.

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions, such as sine, cosine, and tangent. Solving them involves finding unknown angles that satisfy the equation. In this exercise, we dealt with an equation that modeled the force exerted on back muscles.

The equation given was:

• \[F = \frac{0.6 W \sin \left(\theta+90^{\circ}\right)}{\sin 12^{\circ}}\]

One of the tasks was to approximate the force equation using an identity. This turned our original sine-based equation into an easier-to-understand cosine-based approximation:

• \[F \approx 2.9 W \cos \theta\]

When solving trigonometric equations, being able to recognize and apply identities can drastically simplify the problem-solving process. This approach not only aids in finding solutions but also helps in understanding the nature of the solutions and their practical implications.

Problem Solving

Problem-solving with trigonometry often involves the sequential application of mathematical principles and identities to arrive at a solution. This exercise gave us a real-world scenario where an analytical approach was required to find the desired values.

To solve such problems effectively:

• Start by identifying known values and substituting them into the given formula.
• Use trigonometric identities to simplify the equation if necessary. This often involves transforming between sine and cosine, as shown in this problem.
• Then, solve the equation to find the numeric values or to express in simpler forms.

We also explored maximizing a trigonometric function. The key was recognizing that

• \(\cos \theta\) reaches its maximum when \(\theta = 0^\circ\)

Through these steps, problem-solving with trigonometry becomes an organized process where each stage builds on the previous one to reach a logical conclusion. Remember, practice is essential in mastering the art of applying these principles effectively.