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Chapter 6: Problem 22

Simplify each of the trigonometric expressions. $$\sec (-x) \tan (-x) \cos (-x)$$

Short Answer

Expert verified
The simplified expression is \(-\tan(x)\).

Step by step solution

01

Use Trigonometric Identities for Negative Angles

The trigonometric identities for negative angles tell us that \(\sec(-x) = \sec(x)\), \(\tan(-x) = -\tan(x)\), and \(\cos(-x) = \cos(x)\). Let's substitute these identities into the expression: \[\sec(-x) \tan(-x) \cos(-x) = \sec(x)(-\tan(x))\cos(x).\]
02

Simplify the Expression

Take the expression from Step 1: \[\sec(x)(-\tan(x))\cos(x).\]We know that \(\sec(x) = \frac{1}{\cos(x)}\). Substitute \(\sec(x)\) with \(\frac{1}{\cos(x)}\):\[\left(\frac{1}{\cos(x)}\right)(-\tan(x))\cos(x).\]Notice that \(\frac{1}{\cos(x)}\cdot\cos(x) = 1\). This simplifies the expression to:\[-\tan(x).\]

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Key Concepts

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

Negative Angles
When we talk about negative angles, we refer to angles measured in a clockwise direction from the positive x-axis. In trigonometry, negative angles can cause a flip in the direction or sign of certain trigonometric functions. This is why knowing the identities for such angles is helpful.
  • When dealing with negative angles, the secant function remains positive, so \( \sec(-x) = \sec(x) \).
  • For the tangent function, the sign changes, meaning \( \tan(-x) = -\tan(x) \).
  • The cosine function is unaffected by the angle direction, hence \( \cos(-x) = \cos(x) \).
By using these identities, you can simplify trigonometric expressions involving negative angles more easily. Recognize that these changes reflect the symmetrical properties of the unit circle, helping to understand the practical aspect of the sign changes.
Secant Function
The secant function, denoted as \( \sec(x) \), is one of the six trigonometric functions and is primarily known as the reciprocal of the cosine function. This means:\[\sec(x) = \frac{1}{\cos(x)}.\]Understanding the secant function can help simplify expressions in trigonometry. Since it's derived from the cosine, it can run into issues when \( \cos(x) = 0 \), as the function becomes undefined.
  • The secant function is periodic, with a period of \( 2\pi \), like most trigonometric functions.
  • In general, secant gives a ratio related to the opposite of the x-coordinate on the unit circle.
  • Since \( \cos(-x) = \cos(x) \), \( \sec(-x) = \sec(x) \) which shows that secant is even.
Secant plays a crucial role in transformations and can often be simplified using other identities in trigonometric contexts.
Tangent Function
The tangent function, symbolized as \( \tan(x) \), is another fundamental trigonometric function. It represents the ratio of the sine to the cosine:\[\tan(x) = \frac{\sin(x)}{\cos(x)}.\]The behavior and identities of the tangent function are quite unique, primarily due to its period and undefined values.
  • It has a principal period of \( \pi \), meaning its values repeat every \( \pi \) units.
  • The tangent function is odd鈥攊t changes sign with negative angles: \( \tan(-x) = -\tan(x) \).
  • It is undefined wherever \( \cos(x) = 0 \), commonly at odd multiples of \( \frac{\pi}{2} \).
Tangent often appears in slope calculations in real-world functions or as a ratio in triangles. Knowing the properties and these fundamental identities makes solving problems involving tangent more predictable and logical.

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Most popular questions from this chapter

Solve each trigonometric equation on \(0^{\circ} \leq \theta<360^{\circ} .\) Express solutions in degrees and round to two decimal places. $$-\frac{1}{4} \sec ^{2}\left(\frac{x}{8}\right)+\sin ^{2}\left(\frac{x}{8}\right)=0$$

Explain the mistake that is made. $$\text { Solve } \sqrt{2+\sin \theta}=\sin \theta \text { on } 0 \leq \theta \leq 2 \pi$$ Solution: Square both sides. \(2+\sin \theta=\sin ^{2} \theta\). Gather all terms to one side. \(\quad \sin ^{2} \theta-\sin \theta-2=0\). Factor. \(\quad(\sin \theta-2)(\sin \theta+1)=0\). Set each factor equal to zero. \(\sin \theta-2=0\) or \(\sin \theta+1=0\). Solve for \(\sin \theta . \quad \sin \theta=2 \quad\) or \(\quad \sin \theta=-1\). Solve \(\sin \theta=2\) for \(\theta . \quad\) no solution. Solve \(\sin \theta=-1\) for \(\theta . \quad \theta=\frac{3 \pi}{2}\). This is incorrect. What mistake was made?

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$\frac{1}{4} \sec (2 x)=\sin (2 x)$$

Although in general the statement \(\sin (A-B)=\sin A-\sin B\) is not true, it is true for some values. Determine some values of \(A\) and \(B\) that make this statement true.

Determine whether each statement is true or false. $$\sin \left(\frac{\pi}{2}\right)=\sin \left(\frac{\pi}{3}\right)+\sin \left(\frac{\pi}{6}\right)$$
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