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Chapter 10: Problem 97

Use a calculator to find each value. $$\cos \left(\tan ^{-1} 0.5\right)$$

Short Answer

Expert verified
\( \cos(\tan^{-1}(0.5)) \approx 0.894427 \)

Step by step solution

01

Understand the Problem

We need to find the value of \( \cos(\tan^{-1}(0.5)) \). This means we first find the angle \( \theta \) whose tangent is 0.5, then find the cosine of that angle.
02

Find the Angle \( \theta \)

Use the inverse tangent function, \( \tan^{-1} \), to find the angle \( \theta \). In mathematical terms, \( \theta = \tan^{-1}(0.5) \). Use a calculator to compute this value.\[ \theta = \tan^{-1}(0.5) \approx 0.463648 \text{ radians} \]
03

Find Cosine of \( \theta \)

Now that we have \( \theta \approx 0.463648 \) radians, the next step is to find \( \cos(\theta) \). Use the calculator to find the cosine of the angle \( \theta \).\[ \cos(0.463648) \approx 0.894427 \]
04

Final Result

The value of \( \cos(\tan^{-1}(0.5)) \) is approximately 0.894427. This is the cosine of the angle whose tangent is 0.5.

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Key Concepts

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

Inverse Trigonometric Functions
Inverse trigonometric functions are the opposite of the regular trigonometric functions. They help us find the angle when we know the value of a trigonometric ratio like sine, cosine, or tangent. In our exercise, we use the inverse tangent function, written as \( \tan^{-1} \), to calculate the angle whose tangent is \( 0.5 \). This function gives us an angle \( \theta \), and it's expressed in radians or degrees.
  • The \( \tan^{-1} \) function is known as arctangent.
  • The result helps us determine angles from ratios.
  • These angles are crucial for finding other trigonometric values, like using \( \cos(\theta) \) after finding \( \theta \).

Inverse trigonometric functions are essential tools in trigonometry, especially in solving triangles and modeling periodic phenomena in various fields.
Calculators
Calculators make finding trigonometric values simple and fast. Without them, solving problems involving trigonometric functions would take much longer. In our solution, the calculator helps us execute two vital steps: determining \( \theta = \tan^{-1}(0.5) \) and calculating \( \cos(\theta) \).
Firstly, it's essential to know whether your calculator is set to radians or degrees (more on this in the next section).
  • Enter \( 0.5 \) and use the \( \tan^{-1} \) button to find the angle \( \theta \).
  • Use this angle to calculate the cosine value by pressing the \( \cos \) button.

To ensure accuracy, check the calculator's mode and revisit the calculation if discrepancies arise. Calculators are practical tools that simplify complex mathematical operations, making solving trigonometric equations much more straightforward.
Radian Measure
Radians are a way to express angles and are essential in trigonometry and calculus. A full circle is \( 2\pi \) radians, and this measurement becomes particularly important when using trigonometric functions in scientific and math-based calculations. When computing \( \tan^{-1}(0.5) \), our resulting angle \( \theta \) is commonly expressed in radians, approximately \( 0.463648 \) radians.
  • Radians provide a mathematical standard that links coordinates and calculations directly with circle geometry.
  • Most calculators allow users to switch between degree and radian modes for versatility.
  • Understanding radians is crucial for interpreting results and ensuring calculations maintain their intended values.

When tasked with finding trigonometric values, using radian measure keeps calculations coherent and aligns with common practices in higher-level math.

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Most popular questions from this chapter

Verify that each equation is an identity. $$\frac{\tan (A+B)-\tan B}{1+\tan (A+B) \tan B}=\tan A$$

Solve each problem. Hearing Beats in Music Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. 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 the slight variation in the frequency. This phenomenon can be seen on a graphing calculator. (a) Consider two tones with frequencies of 220 and \(223 \mathrm{Hz}\) and pressures \(P_{1}=0.005 \sin 440 \pi t \quad\) and \(\quad P_{2}=0.005 \sin 446 \pi t\) respectively. A graph of \(P_{1}+P_{2}\) as \(Y_{3}\) felt by an eardrum over the 1 -second interval \([0.15,1.15]\) is shown here. How many beats are there in 1 second? (Graph can't copy) (b) Repeat part (a) with frequencies of 220 and 216 (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given.

Solve each problem. Back Stress If a person bends at the waist with a straight back, making an angle of \(\theta\) degrees with the horizontal, then the force \(F\) exerted on the back muscles can be modeled by the equation $$F=\frac{0.6 W \sin \left(\theta+90^{\circ}\right)}{\sin 12^{\circ}}$$ where \(W\) is the weight of the person. (Source: Metcalf, H., Topics in Classical Biophysics, Prentice- Hall.) (a) Calculate \(F\) when \(W=170\) pounds and \(\theta=30^{\circ}\) (b) Use an identity to show that \(F\) is approximately equal to \(2.9 \mathrm{W} \cos \theta\) (c) For what value of \(\theta\) is \(F\) maximum?

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\sin x+\sin 3 x=\cos x$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos \frac{\theta}{2}=1$$
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