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Chapter 7: Problem 67

Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression and all functions are of \(\theta\) only. $$(1-\cos \theta)(1+\sec \theta)$$

Short Answer

Expert verified
\( \sec \theta - \cos \theta \)

Step by step solution

01

- Write in terms of sine and cosine

Express \(\sec \theta\) in terms of \(\cos \theta\). \(\sec \theta = \frac{1}{\cos \theta}\). The expression becomes \( (1 - \cos \theta) \left(1 + \frac{1}{\cos \theta}\right) \).
02

- Simplify the inside of the parentheses

Combine the terms inside the second parenthesis: \(1 + \frac{1}{\cos \theta} = \frac{\cos \theta + 1}{\cos \theta}\). The expression now is \( (1 - \cos \theta) \cdot \frac{\cos \theta + 1}{\cos \theta} \).
03

- Multiply the expressions

Distribute \(1-\cos \theta\) across \(\frac{\cos \theta + 1}{\cos \theta}\): \( (1 - \cos \theta) \cdot \frac{\cos \theta + 1}{\cos \theta} = (1-\cos \theta) \cdot \frac{\cos \theta}{\cos \theta} + (1-\cos \theta) \cdot \frac{1}{\cos \theta} \).
04

- Simplify each term

Simplify each term separately: \( (1-\cos \theta) \cdot \frac{\cos \theta}{\cos \theta} = 1-\cos \theta \). For the second term: \( (1-\cos \theta) \cdot \frac{1}{\cos \theta} = \frac{1-\cos \theta}{\cos \theta}\).
05

- Combine the simplified terms

Combine the simplified terms from Step 4: \( 1 - \cos \theta + \frac{1 - \cos \theta}{\cos \theta} \). Notice that \(\frac{1 - \cos \theta}{\cos \theta} = \frac{1}{\cos \theta} - 1 = \sec \theta - 1 \). The final expression is \( 1 - \cos \theta + \sec \theta - 1 = \sec \theta - \cos \theta \).

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Key Concepts

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

sine and cosine
Understanding the basic trigonometric functions, sine and cosine, is crucial for solving trigonometric problems. These functions represent the ratio of sides in a right-angled triangle. For a given angle \( \theta \):
  • \textbf{sine} (abbreviated as \( \sin \)): Represents the ratio of the length of the side opposite to the angle to the hypotenuse.
  • \textbf{cosine} (abbreviated as \( \cos \)): Represents the ratio of the length of the adjacent side to the angle to the hypotenuse.
Trigonometric identities often transform one trigonometric function into another. For instance, \( \sec \theta = \frac{1}{\cos \theta} \). These identities simplify complex expressions and are fundamental in calculus and beyond.

In our exercise, the expressions were initially given in terms of \( \sec \theta \). By converting \( \sec \theta \) into its cosine form, we set up the basics for further simplifications leading to the desired form.
trigonometric simplification
Trigonometric simplification involves combining and reducing trigonometric expressions. The steps can include:
  • Converting all trigonometric functions to sine and cosine to unify the expression.
  • Finding common denominators to combine fractions.
  • Using basic algebraic principles to multiply and distribute terms.
For example, in the given exercise:

The initial transformation is necessary to express everything in terms of \( \cos \theta \). Following that, we manipulate the fractions by combining, distributing, and simplifying to ensure no quotients are present, as seen in steps 3 and 4. Breaking down these processes into manageable steps allows for a clearer and more systematic approach to trigonometric simplification.
precalculus
Precalculus is the bridge between algebra and calculus, covering essential concepts like trigonometry. Emphasizing operations with trigonometric identities is a core part of precalculus.

In the examined exercise:

  • We started with a trigonometric expression: \( (1-\cos \theta)(1+\sec \theta) \).
  • We converted \( \sec \theta \) and simplified the terms step-by-step.
This multilayered process employed fundamental precalculus skills, demonstrating the link between algebraic manipulation and deeper trigonometric principles.

By mastering these steps in precalculus, students lay a solid foundation for tackling more advanced topics in calculus, ensuring they understand how to transform and simplify expressions systematically and confidently.

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Most popular questions from this chapter

Verify that each equation is an identity. $$\cot 4 \theta=\frac{1-\tan ^{2} 2 \theta}{2 \tan 2 \theta}$$

Solve each problem. When a musical instrument creates a tone of \(110 \mathrm{Hz}\). it also creates tones at \(220,330,440,550,660, \ldots\) Hz. A small speaker cannot reproduce the \(110-\mathrm{Hz}\) vibration but it can reproduce the higher frequencies, which are the upper harmonics. The low tones can still be heard because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is \(110 \mathrm{Hz}\), and this difference tone will be heard by a listener. (Source: Benade, A.. Fundamentals of Musical Acoustics, Dover Publications.) (a) We can model this phenomenon using a graphing calculator. In the window \([0,0.03]\) by \([-1,1],\) graph the upper harmonics represented by the pressure $$ P=\frac{1}{2} \sin [2 \pi(220) t]+\frac{1}{3} \sin [2 \pi(330) t]+\frac{1}{4} \sin [2 \pi(440) t] $$ (b) Estimate all \(t\) -coordinates where \(P\) is maximum. (c) What does a person hear in addition to the frequencies of \(220,330,\) and \(440 \mathrm{Hz} ?\) (d) Graph the pressure produced by a speaker that can vibrate at \(110 \mathrm{Hz}\) and above.

(This discussion applies to functions of both angles and real numbers.) Consider the following. $$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ} \cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta \\\&=-\cos \theta\end{aligned}$$ \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone. Another example of a reduction formula is \(\cos \left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for quickly determining a reduction formula for a trigonometric function \(f\) of the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small positive angle in order to determine the quadrant in which \(Q \pm \theta\) will lie. Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that \(\operatorname{sign}, f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Use these ideas to write reduction formulas for each of the following. $$\tan \left(270^{\circ}-\theta\right)$$

Use a calculator to find each value. Give answers as real numbers. $$\sin \left(\cos ^{-1} 0.25\right)$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2 \sqrt{3} \sin \frac{x}{2}=3$$
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