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Chapter 7: Problem 64

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. $$\sin \theta \sec \theta$$

Short Answer

Expert verified
The expression simplifies to \( \tan \theta \).

Step by step solution

01

Express secant in terms of cosine

Recall that the secant function is the reciprocal of the cosine function. Therefore, \ \[ \sec \theta = \frac{1}{\cos \theta} \].
02

Substitute secant into the original expression

Substitute the expression for secant into the original equation: \[ \sin \theta \sec \theta = \sin \theta \cdot \frac{1}{\cos \theta} \].
03

Simplify the expression

Simplify the expression by multiplying the terms: \[ \sin \theta \cdot \frac{1}{\cos \theta} = \frac{\sin \theta}{\cos \theta} \].
04

Use the definition of tangent

Recall that \( \frac{\sin \theta}{\cos \theta} = \tan \theta \). Therefore, the expression simplifies to \( \tan \theta \).

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Key Concepts

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

sine and cosine
Sine and cosine are fundamental trigonometric functions that relate the angles of a triangle to the lengths of its sides.
The sine of an angle \theta is defined as the ratio of the length of the opposite side to the hypotenuse. In a right-angled triangle, this can be written as: \( \text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
Cosine, on the other hand, is the ratio of the adjacent side to the hypotenuse: \( \text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
These functions are foundational in trigonometry and have a range of applications from geometry to physics.
secant function
The secant function, or \sec \theta, is less commonly discussed than sine and cosine but is very important in trigonometry.
Secant is defined as the reciprocal of the cosine function: \( \text{sec} \theta = \frac{1}{\text{cos} \theta} \).
This means that wherever you see \sec \theta, you can replace it with \( \frac{1}{\text{cos} \theta} \). This placement helps simplify equations by converting all terms into sine and cosine, which are easier to work with.
tangent definition
Tangent is another primary trigonometric function.
It is defined as the ratio of sine to cosine: \( \text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \).
In right-angled triangles, this is the ratio of the opposite side to the adjacent side.
By understanding and using the definition of tangent, you can simplify various trigonometric expressions just like in the exercise where \( \frac{\text{sin} \theta}{\text{cos} \theta} = \text{tan} \theta \).
trigonometric simplification
Trigonometric simplification involves rewriting functions and expressions using fundamental identities for easier computation.
  • Start by expressing all functions in terms of sine and cosine whenever possible.
  • Use reciprocal identities such as \( \text{sec} \theta = \frac{1}{\text{cos} \theta} \) to eliminate fractions.
  • Apply quotient identities like \( \text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \) to simplify the expression further.
In the given exercise, this was done step-by-step, first changing \( \text{sec} \theta \) into \( \frac{1}{\text{cos} \theta} \), then simplifying and using the tangent definition to reach \text{tan} \theta.
These strategies ensure that no quotients remain in the final expression, making it more straightforward and elegant.

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Most popular questions from this chapter

Solve each problem. Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. Beats occur when two tones vary in frequency by only a few hertz. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of this slight variation in the frequency. This phenomenon can be seen using a graphing calculator. (Source: Pierce, \(\mathrm{J}\)., The Science of Musical Sound, Scientific American Books.) (a) Consider the two tones with frequencies of \(220 \mathrm{Hz}\) and \(223 \mathrm{Hz}\) and pressures \(P_{1}=0.005 \sin 440 \pi t\) and \(P_{2}=0.005\) sin \(446 \pi t,\) respectively. Graph the pressure \(P=P_{1}+P_{2}\) felt by an eardrum over the 1 -sec interval \([0.15,1.15] .\) How many beats are there in 1 sec? (b) Repeat part (a) with frequencies of 220 and \(216 \mathrm{Hz}\) (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given.

Solve each equation in \(x\) for exact solutions over the interval \([0,2 \pi)\) and each equation in \(\theta\) for exact solutions over the interval 360 $$\tan 4 x=0$$

Verify that each equation is an identity. $$\frac{2}{1+\cos x}-\tan ^{2} \frac{x}{2}=1$$

A coil of wire rotating in a magnetic field induces a voltage $$E=20 \sin \left(\frac{\pi t}{4}-\frac{\pi}{2}\right)$$ Use an identity from this section to express this in terms of \(\cos \frac{\pi t}{4}\).

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically. $$\frac{\tan \frac{x}{2}+\cot \frac{x}{2}}{\cot \frac{x}{2}-\tan \frac{x}{2}}$$
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