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Chapter 1: Problem 9

Use a calculator to determine four-digit decimal approximations for each of the following. (a) \(\csc (1)\) (b) \(\tan \left(\frac{12 \pi}{5}\right)\) (c) \(\cot (5)\) (d) \(\sec \left(\frac{13 \pi}{8}\right)\) (e) \(\sin ^{2}(5.5)\) (f) \(1+\tan ^{2}(2)\) (g) \(\sec ^{2}(2)\)

Short Answer

Expert verified
(a) \(\csc(1) \approx 1.1884\) (b) \(\tan\left(\frac{12\pi}{5}\right) \approx 0.7265\) (c) \(\cot(5) \approx -0.2952\) (d) \(\sec\left(\frac{13\pi}{8}\right) \approx -1.0823\) (e) \(\sin^2(5.5) \approx 0.7081\) (f) \(1+\tan^2(2) \approx 3.4265\) (g) \(\sec^2(2) \approx 1.9198\)

Step by step solution

01

(a) Calculating \(\csc(1)\)

First, we need to find the sine of 1 using the calculator, which should be in radians mode. After that, take the reciprocal of the sine value, which will give us the value of \(\csc(1)\). Then, round the result to four decimal places.
02

(b) Calculating \(\tan\left(\frac{12\pi}{5}\right)\)

In your calculator, find the tangent of \(\frac{12\pi}{5}\) radians. Make sure your calculator is in radians mode. Round the result to four decimal places.
03

(c) Calculating \(\cot(5)\)

First, find the tangent of 5 radians using your calculator, making sure it's in radians mode. Then, take the reciprocal of the tangent value, which will give us the value of \(\cot(5)\). Round the result to four decimal places.
04

(d) Calculating \(\sec\left(\frac{13\pi}{8}\right)\)

Find the cosine of \(\frac{13\pi}{8}\) radians using your calculator, making sure it's in radians mode. Afterwards, take the reciprocal of the cosine value, which will give us the value of \(\sec\left(\frac{13\pi}{8}\right)\). Round the result to four decimal places.
05

(e) Calculating \(\sin^2(5.5)\)

Find the sine of 5.5 radians using your calculator. Make sure your calculator is in radians mode. Then, square the sine value, giving you the value of \(\sin^2(5.5)\). Round the result to four decimal places.
06

(f) Calculating \(1+\tan^2(2)\)

First, find the tangent of 2 radians using your calculator. Once your calculator is in radians mode, square the tangent value. Then, add 1 to the squared tangent value, giving you the value of \(1+\tan^2(2)\). Round the result to four decimal places.
07

(g) Calculating \(\sec^2(2)\)

First, find the cosine of 2 radians using your calculator, making sure it's in radians mode. Then, take the reciprocal of the cosine value. After that, square the reciprocal cosine value, giving you the value of \(\sec^2(2)\). Round the result to four decimal places.

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Key Concepts

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

Cosecant
The cosecant function, denoted as \(\csc(\theta)\), is the reciprocal of the sine function. It is defined as:
  • \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
This function helps in finding the hypotenuse-to-opposite side ratio in a right-angled triangle. To calculate \(\csc(1)\), follow these steps:
  • Ensure your calculator is set to radians mode. This is crucial since most trigonometric calculations, especially in calculus, use radians.
  • First, find \(\sin(1)\) using the calculator.
  • Take the reciprocal of this value to find \(\csc(1)\).
This provides a deeper understanding of how cosecant works and why it鈥檚 useful in trigonometric calculations.
Secant
The secant function, represented by \(\sec(\theta)\), is the reciprocal of the cosine function.
  • \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
This function measures the hypotenuse-to-adjacent side ratio in a triangle. Here鈥檚 how you determine \(\sec\left(\frac{13\pi}{8}\right)\):
  • Set your calculator to radians mode.
  • Calculate \(\cos\left(\frac{13\pi}{8}\right)\).
  • Take its reciprocal to find \(\sec\left(\frac{13\pi}{8}\right)\).
Understanding secant is essential for solving problems where you need to relate angles to sides differently from sine or cosine.
Cotangent
The cotangent function, denoted \(\cot(\theta)\), is the reciprocal of the tangent function.
  • \(\cot(\theta) = \frac{1}{\tan(\theta)}\)
It represents the adjacent-to-opposite side ratio in a triangle. To calculate \(\cot(5)\):
  • Ensure your calculator is in radians mode.
  • Find \(\tan(5)\) using your calculator.
  • Take the reciprocal of this result to get \(\cot(5)\).
Understanding cotangent helps in analyzing angles where the tangent is not directly applicable.
Radians Mode Calculation
Radians are a way of measuring angles based on the radius of a circle. Here鈥檚 why they are important:
  • Radians create a natural connection between the arc length of a circle and the angle itself.
For calculations:
  • Ensure your calculator is in radians mode; this is critical for most trigonometry problems.
  • Radians make integrations and differentiations simpler in calculus since they naturally fit the functions.
When calculating trigonometric functions like sine, cosine, or tangent, the mode of your calculator should match the unit of the angle you are working with. This ensures correctness and consistency in results.

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Most popular questions from this chapter

(a) What are the possible values of \(\cos (t)\) if it is known that \(\sin (t)=\frac{3}{5} ?\) (b) What are the possible values of \(\cos (t)\) if it is known that \(\sin (t)=\frac{3}{5}\) and the terminal point of \(t\) is in the second quadrant? (c) What is the value of \(\sin (t)\) if it is known that \(\cos (t)=-\frac{2}{3}\) and the terminal point of \(t\) is in the third quadrant?

Determine the exact value for each of the following expressions and then use a calculator to check the result. For example, $$\cos (0)+\sin \left(\frac{\pi}{3}\right)=1+\frac{\sqrt{3}}{2} \approx 1.8660$$ (a) \(\cos ^{2}\left(\frac{\pi}{6}\right)\) (b) \(2 \sin ^{2}\left(\frac{\pi}{4}\right)+\cos (\pi)\) (c) \(\frac{\cos \left(\frac{\pi}{6}\right)}{\sin \left(\frac{\pi}{6}\right)}\) (d) \(3 \sin \left(\frac{\pi}{2}\right)+\cos \left(\frac{\pi}{4}\right)\)

A small pulley with a radius of 10 centimeters inches is connected by a belt to a larger pulley with a radius of 24 centimeters inches (See Figure 1.16 ). The larger pulley is connected to a motor that causes it to rotate counterclockwise at a rate of \(75 \mathrm{rpm}\) (revolutions per minute). Because the two pulleys are connected by the belt, the smaller pulley also rotates in the counterclockwise direction. (a) Determine the angular velocity of the larger pulley in radians per minute. (b) Determine the linear velocity of the rim of the large pulley in inches per minute. (c) What is the linear velocity of the rim of the smaller pulley? Explain. (d) Find the angular velocity of the smaller pulley in radians per second. (e) How many revolutions per minute is the smaller pulley turning?

A small pulley with a radius of 3 inches is connected by a belt to a larger pulley with a radius of 7.5 inches (See Figure 1.16 ). The smaller pulley is connected to a motor that causes it to rotate counterclockwise at a rate of 120 rpm (revolutions per minute). Because the two pulleys are connected by the belt, the larger pulley also rotates in the counterclockwise direction. (a) Determine the angular velocity of the smaller pulley in radians per minute. (b) Determine the linear velocity of the rim of the smaller pulley in inches per minute. (c) What is the linear velocity of the rim of the larger pulley? Explain. (d) Find the angular velocity of the larger pulley in radians per minute. (e) How many revolutions per minute is the larger pulley turning?

Determine the quadrant in which the terminal point of each arc lies based on the given information. (a) \(\cos (x)>0\) and \(\tan (x)<0\) (b) \(\tan (x)>0\) and \(\csc (x)<0\) (c) \(\cot (x)>0\) and \(\sec (x)>0\) (d) \(\sin (x)<0\) and \(\sec (x)>0\) (e) \(\sec (x)<0\) and \(\csc (x)>0\) (f) \(\sin (x)<0\) and \(\cot (x)>0\)
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