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Chapter 7: Problem 70

The following equation occurs in the study of mechanics: $$\sin \theta=\frac{I_{1} \cos \phi}{\sqrt{\left(I_{1} \cos \phi\right)^{2}+\left(I_{2} \sin \phi\right)^{2}}}$$ It can happen that \(I_{1}=I_{2} .\) Assuming that this happens, simplify the equation.

Short Answer

Expert verified
\(\sin \theta = \cos \phi\).

Step by step solution

01

Identify Given Values

Recognize that we are given the condition where \(I_1 = I_2\). This simplifies our equation since both values are the same.
02

Substitute Given Values

Substitute \(I_1 = I_2 = I\) into the equation. Our original equation \(\frac{I_{1} \cos \phi}{\sqrt{\left(I_{1} \cos \phi\right)^{2} + \left(I_{2} \sin \phi\right)^{2}}} \) will now be written as \( \sin \theta = \frac{I \cos \phi}{\sqrt{\left(I \cos \phi \right)^{2} + \left(I \sin \phi\right)^{2}}} \).
03

Simplify the Denominator

Since \(I_1 = I_2 = I\), rewrite the denominator: \[\left( I \cos \phi\right)^{2} + \left( I \sin \phi \right)^{2} = I^{2}\cos^{2} \phi + I^{2}\sin^{2}\phi\]. Here, we can factor out the common term \(I^2\): \[ I^{2} \left( \cos^{2} \phi + \sin^{2} \phi \right) \]. Recall that \(\cos^{2} \phi + \sin^{2} \phi = 1 \), so the equation simplifies to: \( I^{2} \left( 1 \right) = I^{2} \).
04

Simplify the Fraction

Replace the denominator with \(I^2\) in our simplified equation \( \sin \theta = \frac{I \cos \phi}{\sqrt{I^2}} \), which simplifies further to \( \sin \theta = \frac{I \cos \phi}{I} \). This reduces further since \( I \) cancels out: \( \sin \theta = \cos \phi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are fundamental tools in simplifying trigonometric equations. We often use these identities to transform and reduce complex expressions. In this exercise, one essential identity is \(\text{cos}^2 \phi + \text{sin}^2 \phi = 1\). This relationship is crucial because it allows us to simplify the given equation substantially. Understanding and memorizing these identities can help you recognize patterns and simplify expressions effectively. Other key identities are:
  • Pythagorean identities: \(\text{tan}^2 \theta + 1 = \text{sec}^2 \theta\) and \(1 + \text{cot}^2 \theta = \text{csc}^2 \theta\)
  • Angle-sum identities: \(\text{sin}(a + b) = \text{sin} a \text{cos} b + \text{cos} a \text{sin} b\) and \(\text{cos}(a + b) = \text{cos} a \text{cos} b - \text{sin} a \text{sin} b\)
Using these identities can simplify complex trigonometric problems and make algebraic manipulation more manageable.
Mechanics
Mechanics often involves the use of trigonometric functions to resolve forces and analyze motion. Equations such as the one in this exercise can appear in studies involving rotational dynamics or analyzing mechanical systems. Simplifying these equations makes it easier to understand and solve problems related to physical situations. When you encounter an equation like \(\text{sin} \theta = \frac{I_1 \text{cos} \phi}{\text{sqrt}((I_1 \text{cos} \phi)^2 + (I_2 \text{sin} \phi)^2)}\), recognizing that the trigonometric components might represent geometric relationships or force vectors is essential. Analyzing these components helps break down the physical situation and solve complex mechanical problems.
Algebraic Manipulation
Algebraic manipulation is a critical skill in simplifying equations. In this exercise, it involves substituting given values and making use of identities to reduce the equation. Here鈥檚 how it's done step-by-step:
  • Substitute Known Values: Initially, replace \(I_1 = I_2 = I\) to simplify the terms.
  • Combine Like Terms: In the denominator, factor out \(I^2\) and use the identity \(\text{cos}^2 \phi + \text{sin}^2 \phi = 1\).
  • Simplify the Fraction: Replace the denominator by simplifying \(I^2 \times 1 = I^2\). Finally, cancel out the \(I\) terms to reduce expressions.
These steps show that meticulous algebraic manipulation can simplify terms significantly and clarify complex equations. Being proficient in these techniques is essential for solving algebraic equations across various domains, including trigonometry and mechanics.

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Most popular questions from this chapter

Angles Between Lines. One of the identities gives an easy way to find an angle formed by two lines. Consider two lines with equations \(l_{1}: y=m_{1} x+b_{1}\) and \(l_{2}: y=m_{2} x+b_{2}\) (GRAPH CANNOT COPY) The slopes \(m_{1}\) and \(m_{2}\) are the tangents of the angles \(\theta_{1}\) and \(\theta_{2}\) that the lines form with the positive direction of the \(x\) -axis. Thus we have \(m_{1}=\tan \theta_{1}\) and \(m_{2}=\tan \theta_{2} .\) To find the measure of \(\theta_{2}-\theta_{1},\) or \(\phi,\) we proceed as follows: This formula also holds when the lines are taken in the reverse order. When \(\phi\) is acute, tan \(\phi\) will be positive. When \(\phi\) is obtuse, tan \(\phi\) will be negative. Find the measure of the angle from \(l_{1}\) to \(l_{2}\) $$\begin{aligned} &l_{1}: 2 x+y-4=0\\\ &l_{2}: y-2 x+5=0 \end{aligned}$$

First write each of the following as a trigonometric function of a single angle. Then evaluate. $$\frac{\tan 20^{\circ}+\tan 32^{\circ}}{1-\tan 20^{\circ} \tan 32^{\circ}}$$

Evaluate. \(\cos \left(\tan ^{-1} \frac{\sqrt{3}}{4}\right)\)

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { linear speed } & \text { congruent } \\ \text { angular speed } & \text { circular } \\ \text { angle of elevation } & \text { periodic } \\ \text { angle of depression } & \text { period } \\ \text { complementary } & \text { amplitude } \\ \text { supplementary } & \text { quadrantal } \\ \text { similar } & \text { radian measure }\end{array}$$ Trigonometric functions with domains composed of real numbers are called ____________ functions.

Derive the identity. Check using a graphing calculator. $$\tan \left(x+\frac{\pi}{4}\right)=\frac{1+\tan x}{1-\tan x}$$
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