Problem 15 - Algebra 2 and Trigonometry

Question

An engineer would like to model a piece for a factory machine on his computer. As shown in the figure, the machine consists of a link fixed to a circle at point \(A\) . The other end of the link is fixed to a slider at point B. As the circle rotates, point \(B\) slides back and forth between the two ends of the slider \((C \text { and } D)\) . The movement is restricted so that \(\theta\) , the measure of \(\angle A O D,\) is in the interval \(-45^{\circ} \leq \theta \leq 45^{\circ} .\) The motion of point \(B\) can be described mathematically by the formula $$ C B=r(\cos \theta-1)+\sqrt{l^{2}-r^{2} \sin ^{2} \theta} $$ where \(r\) is the radius of the circle and \(l\) is the length of the link. Both the radius of the circle and the length of the link are 2 inches. a. Find the exact value of \(C B\) when: \((1) \theta=30^{\circ}(2) \theta=45^{\circ} .\) b. Find the exact value(s) of \(\theta\) when \(C B=2\) inches. c. Find, to the nearest hundredth of a degree, the value(s) of \(\theta\) when \(C B=1.5\) inches.

Step-by-Step Solution

Answer

a. (1) \( CB = 2\sqrt{3} - 2 \), (2) \( CB = 2\sqrt{2} - 2 \); b. \( \theta = 0^\circ \); c. \( \theta \approx 29^\circ \).

Step 1: Define known variables

We know from the problem that the radius of the circle, \(r\), and the length of the link, \(l\), are both 2 inches. We will substitute these values into the given equation: \( CB = r(\cos \theta - 1) + \sqrt{l^2 - r^2 \sin^2 \theta} \).

Step 2: Calculate CB for \(\theta = 30^\circ\)

Substitute \(\theta = 30^\circ\) and the known values of \(r\) and \(l\) into the equation: \[CB = 2(\cos 30^\circ - 1) + \sqrt{2^2 - 2^2 \sin^2 30^\circ}.\]Simplify using \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\):\[CB = 2 \left(\frac{\sqrt{3}}{2} - 1\right) + \sqrt{4 - 1} = \sqrt{3} - 2 + \sqrt{3} = 2\sqrt{3} - 2.\]

Step 3: Calculate CB for \(\theta = 45^\circ\)

Substitute \(\theta = 45^\circ\) and the known values of \(r\) and \(l\) into the equation:\[CB = 2(\cos 45^\circ - 1) + \sqrt{2^2 - 2^2 \sin^2 45^\circ}.\]Simplify using \(\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\):\[CB = 2 \left(\frac{\sqrt{2}}{2} - 1\right) + \sqrt{4 - 2} = \sqrt{2} - 2 + \sqrt{2} = 2\sqrt{2} - 2.\]

Step 4: Solve for \(\theta\) when \(CB = 2\) inches

Set the equation equal to 2 and solved for \(\theta\):\[2(\cos \theta - 1) + \sqrt{4 - 4\sin^2 \theta} = 2.\]Simplify to: \[2\cos \theta - 2 + \sqrt{4(1 - \sin^2 \theta)} = 2.\]Since \(\sqrt{4(1 - \sin^2 \theta)} = 2\cos \theta\), the equation simplifies directly: \[2\cos \theta = 2,\] which implies \(\cos \theta = 1\). Therefore, \(\theta = 0^\circ\).

Step 5: Solve for \(\theta\) when \(CB = 1.5\) inches

Set the equation equal to 1.5 inches:\[2(\cos \theta - 1) + \sqrt{4 - 4\sin^2 \theta} = 1.5.\]Simplifying gives:\[2\cos \theta - 2 + \sqrt{4 - 4\sin^2 \theta} = 1.5.\]Isolate for \(\cos \theta\):\[2\cos \theta - 2 + 2\cos \theta = 1.5.\]\(4\cos \theta = 3.5\) \[\cos \theta = \frac{3.5}{4} = 0.875,\]Solve for \(\theta\) using a calculator:\(\theta \approx \cos^{-1}(0.875) \approx 28.955^\circ\)Therefore, \(\theta \approx 29^\circ\), since \(\theta\) is constrained to between -45° and 45°, the only solution within these bounds is \(\theta \approx 29^\circ\).

Key Concepts

Understanding the Cosine Function

The cosine function is a fundamental concept in trigonometry. It relates the angle measured in a standard position to the horizontal coordinate of a point on the unit circle. In simpler terms, given an angle \(\theta\), the cosine of this angle tells you how far left or right a point is on a circle with radius one. When \(\theta\) is 0°, the cosine is at its maximum value of 1 because the point is at (1,0) on the unit circle.

• For \(\theta = 90°\), cosine is 0 because at 90 degrees, the point is at the origin's y-axis, (0,1).
• For \(\theta = 180°\), cosine is -1, reflecting the point at (-1,0).
• The cosine function oscillates between 1 and -1 as \(\theta\) varies.

In problems involving engineering mechanics or physics, the cosine function often determines horizontal components of movement or force, seen in this problem by calculating horizontal displacement as part of "CB" for different angles.

Angle Measurement and Its Importance

Angles are typically measured in degrees or radians. In this exercise, angles are constrained between -45° and 45°, showing its importance in limiting the range of motion or forces in practical engineering applications.

• Angles less than \(\theta = 0°\) in this context move the point B to the left.
• Angles greater than \(\theta = 0°\), move point B to the right.

Accurate angle measurement ensures correct application of formulas that involve trigonometric functions. Here, it is essential for determining the precise position of point B's sliding motion, linking rotational motion to linear motion in the physical setup.

Problem Solving with Trigonometric Functions

Using trigonometric formulas such as the one provided establishes a good challenge in problem solving. Solving for \(CB\) when \(\theta\) is known or vice versa involves applying known values, simplifying expressions, and manipulating them mathematically. For known angles like 30° or 45°, the trigonometric functions have well-documented values that simplify the procedure:

• Use of standard trigonometric values, like \(\cos 30° = \frac{\sqrt{3}}{2}\), helps in readily evaluating the expression.
• Understanding identity relationships such as \(\cos^2 \theta + \sin^2 \theta = 1\) aids simplification as seen with solving for \(\sqrt{4 - 4\sin^2 \theta} = 2\cos \theta\).

When calculating unknown angles from known lengths (e.g., \(CB = 2\)), solution steps include setting up equations and solving for \(\theta\) using inverse trigonometric functions. This systematic approach provides insights on the relationships between physical and geometric aspects in engineering designs.