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Chapter 6: Problem 40

Four functions \(S, C, T,\) and \(D\) are defined as follows: \(\left.\begin{array}{l}S(\theta)=\sin \theta \\ C(\theta)=\cos \theta \\\ T(\theta)=\tan \theta \\ D(\theta)=2 \theta\end{array}\right\\} \quad 0^{\circ}<\theta<90^{\circ}\) In each case, use the values to decide if the statement is true or false. A calculator is not required. $$T\left(45^{\circ}\right)-(C \circ D)\left(30^{\circ}\right)>0$$

Short Answer

Expert verified
The statement is true.

Step by step solution

01

Evaluate Function T at 45掳

Given that \( T(\theta) = \tan \theta \), we need to find \( T(45^{\circ}) \). We know that \( \tan 45^{\circ} = 1 \). Therefore, \( T(45^{\circ}) = 1 \).
02

Evaluate Function D at 30掳

Function \( D(\theta) \) is defined as \( D(\theta) = 2\theta \). Thus, substituting \( \theta = 30^{\circ} \), we get \( D(30^{\circ}) = 2 \times 30^{\circ} = 60^{\circ} \).
03

Evaluate Function C at 60掳

Given that \( C(\theta) = \cos \theta \), we need to evaluate \( C(60^{\circ}) \). From trigonometric values, we know that \( \cos 60^{\circ} = \frac{1}{2} \). Therefore, \( C(60^{\circ}) = \frac{1}{2} \).
04

Calculate Expression

We need to calculate \( T(45^{\circ}) - (C \circ D)(30^{\circ}) \). We already determined that \( T(45^{\circ}) = 1 \) and \( (C \circ D)(30^{\circ}) = C(60^{\circ}) = \frac{1}{2} \). Thus, the expression becomes \( 1 - \frac{1}{2} \).
05

Simplify and Determine Truth Value

Simplify the expression: \( 1 - \frac{1}{2} = \frac{1}{2} \). Since \( \frac{1}{2} > 0 \), the statement is true.

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Key Concepts

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

Sine
The sine function is one of the fundamental trigonometric functions used in geometry and trigonometry. It describes the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle. Mathematically, it's represented as:\[\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\]Sine values are important in various applications:
  • Sine of 0掳 is 0.
  • Sine of 30掳 is \(\frac{1}{2}\).
  • Sine of 90掳 is 1.
Understanding the sine function helps us in solving problems related to angles and periodic phenomena like sound waves.
Cosine
The cosine function complements the sine function in trigonometry. It represents the ratio of the length of the adjacent side of the angle to the hypotenuse in a right triangle:\[\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\]Here are some useful cosine values to remember:
  • \(\cos 0^{\circ} = 1\)
  • \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
  • \(\cos 60^{\circ} = \frac{1}{2}\)
  • \(\cos 90^{\circ} = 0\)
Cosine is integral in calculating lengths and angles in triangles, as well as modeling waves and oscillations.
Tangent
The tangent function is another crucial trigonometric function. It can be understood as the ratio between the sine and cosine of an angle, or more practically, the ratio of the opposite side to the adjacent side in a right triangle:\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{Opposite}}{\text{Adjacent}}\]Key properties of the tangent function include:
  • \(\tan 0^{\circ} = 0\)
  • \(\tan 45^{\circ} = 1\)
  • \(\tan 90^{\circ}\) is undefined because \(\cos 90^{\circ} = 0\).
Tangent is very useful for angles that are not commonly found in simple geometric setups and for calculating angles of inclination.
Trigonometric Identities
Trigonometric identities are equations that relate trigonometric functions to one another. These identities simplify complex trigonometric expressions and make calculations more manageable. Some of the most useful identities include:
  • **Pythagorean Identity:** \(\sin^2 \theta + \cos^2 \theta = 1\)
  • **Reciprocal Identities:**
    • \(\csc \theta = \frac{1}{\sin \theta}\)
    • \(\sec \theta = \frac{1}{\cos \theta}\)
    • \(\cot \theta = \frac{1}{\tan \theta}\)
  • **Angle Sum and Difference Identities:**
    • \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)
    • \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)
These identities are tools that help us move from one form of a function to another, greatly simplifying the solving process.

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Most popular questions from this chapter

Use the given information to determine the remaining five trigonometric values. $$\cos \theta=-3 / 5, \quad 180^{\circ}<\theta<270^{\circ}$$

Four functions \(S, C, T,\) and \(D\) are defined as follows: \(\left.\begin{array}{l}S(\theta)=\sin \theta \\ C(\theta)=\cos \theta \\\ T(\theta)=\tan \theta \\ D(\theta)=2 \theta\end{array}\right\\} \quad 0^{\circ}<\theta<90^{\circ}\) In each case, use the values to decide if the statement is true or false. A calculator is not required. $$(T \circ D)\left(15^{\circ}\right)-C\left(30^{\circ}\right)>0$$

Use a calculator to evaluate \(\sec \theta, \csc \theta,\) and cot \(\theta\) for the given value of \(\theta .\) Round the answers to two decimal places. $$5.23$$

Let \(P(x, y)\) denote the point where the terminal side of angle \(\boldsymbol{\theta}\) (in standard position) meets the unit circle (as in Figure 4). Use the given information to evaluate the six trigonometric functions of \(\theta\). $$x=-8 / 15 \text { and } \pi / 2<\theta<\pi$$

This exercise completes the discussion in the text concerning the use of parentheses in calculator work. (a) Use the unit circle definitions to briefly explain (in complete sentences) why \(\sin (\pi / 2)=1\) and \(\sin \pi=0\). (b) Set the calculator to the radian mode and enter the following sequence of keystrokes. $$\begin{array}{|c|c|c|c|c|} \hline \text { sin } & \pi & \div & 2 & \text { ENTER } \\ \hline \end{array}$$ Your calculator will show an output of \(0,\) which, as you know, is not the value of \(\sin (\pi / 2) .\) This is because, in the absence of parentheses, the calculator interprets the sequence of keystrokes \((\sin ) \quad \pi \quad(\div) \quad 2\) as follows: First compute sin \(\pi\), then divide the result by 2 That is, the calculator computes \(0 \div 2,\) which, of course, results in the 0 output. Conclusion: If you want the calculator to compute \(\sin (\pi / 2),\) you must use parentheses and enter the sequence of keystrokes
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