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Chapter 5: Problem 40

Find the approximate value of each expression. Round to four decimal places. $$\cot \left(179.4^{\circ}\right)$$

Short Answer

Expert verified
143.2502

Step by step solution

01

Convert the angle to radians

To find the trigonometric function values, first convert the given angle from degrees to radians. Use the conversion formula: \[ \text{radians} = \text{degrees} \times \frac{\text{蟺}}{180} \] For the angle 179.4掳, the conversion is: \[ 179.4^{\text{掳}} \times \frac{\text{蟺}}{180} = 3.1311 \text{ radians} \]
02

Find the tangent of the angle

Next, compute the tangent of the angle in radians. Use a calculator or software to find \[ \tan(3.1311) \] \[ \tan(3.1311) \thickapprox 0.006982 \]
03

Calculate the cotangent

Now, find the cotangent by taking the reciprocal of the tangent: \[ \text{cot}(179.4^{\text{掳}}) = \frac{1}{\tan(3.1311)} \] \[ \frac{1}{0.006982} \thickapprox 143.2502 \] Round the result to four decimal places, so the answer is approximately 143.2502.

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Key Concepts

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

Angle Conversion to Radians
When working with trigonometric functions, angles must often be converted from degrees to radians. Radians are a unit of angular measure used in mathematics. To convert an angle from degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\text{蟺}}{180} \). This formula involves multiplying the angle in degrees by \( \frac{\text{蟺}}{180} \), since 180 degrees is equal to 蟺 radians. It's crucial to make this conversion for accurate trigonometric calculations because most mathematical functions use radians by default. For example, to convert 179.4掳 to radians, calculate: \( 179.4^{\text{掳}} \times \frac{\text{蟺}}{180} = 3.1311 \) radians.
Trigonometric Functions
Trigonometric functions are mathematical functions related to angles of triangles. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan).
  • **Sine (sin)**: Computes the ratio of the length of the opposite side to the hypotenuse.
  • **Cosine (cos)**: Computes the ratio of the length of the adjacent side to the hypotenuse.
  • **Tangent (tan)**: Computes the ratio of the length of the opposite side to the adjacent side.
In the context of our calculation, we needed to find the tangent of the angle after converting it to radians. Using a calculator, we find: \( \tan(3.1311) \approx 0.006982 \). This value is then used to find the cotangent.
Tangent and Cotangent Relationship
The tangent and cotangent functions are closely related in trigonometry. The cotangent of an angle is the reciprocal of the tangent of the same angle. Mathematically, this relationship is expressed as: \( \text{cot}(x) = \frac{1}{\tan(x)} \).This means if you know the tangent of an angle, you can easily calculate the cotangent by taking its reciprocal. In our previous step, we found that \( \tan(3.1311) \approx 0.006982 \). To find the cotangent, calculate: \( \text{cot}(3.1311) = \frac{1}{0.006982} \approx 143.2502 \).Thus, understanding the relationship between these functions makes our trigonometric calculations straightforward and consistent.

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Most popular questions from this chapter

Find the exact value of each function. a. \(\sin \left(45^{\circ}\right)\) b. \(\cos (-\pi / 4)\)

Determine the period and sketch at least one cycle of the graph of each function. $$y=\cot (2 x-\pi / 2)$$

For the past three years, the manager of The Toggery Shop has observed that revenue reaches a high of about 40,000 dollars in December and a low of about 10,000 dollars in June, and that a graph of the revenue looks like a sinusoid. If the months are numbered 1 through 36 with 1 corresponding to January, then what are the period, amplitude, and phase shift for this sinusoid? What is the vertical translation? Write a formula for the curve and find the approximate revenue for April.

Determine the period and sketch at least one cycle of the graph of each function. $$y=-\tan (x-\pi / 2)$$

Find the approximate value of each expression. Round to four decimal places. $$\sec \left(-0.024^{\circ}\right)$$
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