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Chapter 6: Problem 58

Find the function value. Round to four decimal places. $$\sin 26.1^{\circ}$$

Short Answer

Expert verified
0.4397

Step by step solution

01

Identify the Function

The problem requires finding the sine of a given angle, which is 26.1 degrees. This involves using the trigonometric function \( \sin \theta \).
02

Use a Calculator

To find \( \sin 26.1^{\circ} \, \), input 26.1 degrees into a scientific calculator or use an online trigonometric calculator in degree mode.
03

Calculate the Sine Value

After inputting 26.1 degrees into the calculator, the sine value is approximately 0.4397. We round this value to four decimal places.

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Key Concepts

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

sine function
The sine function is a fundamental concept in trigonometry. It is denoted as \( \sin\theta \) and is used to determine the ratio of the length of the opposite side to the hypotenuse in a right triangle.

In other words, for any angle \( \theta \), \( \sin\theta \) can be thought of as: \[\frac{opposite}{hypotenuse}\]. Here, 'opposite' is the side opposite to angle \( \theta \), and 'hypotenuse' is the longest side of the triangle.

The sine function has values that range from -1 to 1. When you have an angle in degrees, like 26.1 degrees in the exercise, you can easily find \( \sin\theta \) using a calculator or trigonometric tables.
degree mode
When working with angles, it's crucial to ensure your calculator is in the correct mode. In trigonometry, angles can be measured in degrees or radians.

Degree mode means that the angles are measured in degrees, which is a standard way of measuring angles in everyday contexts. For instance, a right angle is 90 degrees.

If your calculator is in radian mode, the results for trigonometric functions will be different, leading to incorrect answers. Always double-check that your calculator is set to degree mode when dealing with angles given in degrees.

For instance, to find \( \sin 26.1^{\circ} \), make sure your calculator is in degree mode.
scientific calculator
A scientific calculator is an essential tool for solving trigonometric problems. It allows you to perform various functions beyond basic arithmetic, including trigonometric functions like sine, cosine, and tangent.

Here's a quick guide to using a scientific calculator for finding \( \sin 26.1^{\circ} \):
  • First, ensure your calculator is in degree mode. There is usually a 'MODE' button to switch between degrees and radians.
  • Next, input '26.1'.
  • Finally, press the \( \sin \) button to find the sine value. Your calculator should display approximately 0.4397.

Remember, always round your answer to the required number of decimal places, in this case, four decimal places.

Using a scientific calculator accurately ensures that you can solve trigonometric problems efficiently and correctly.

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Most popular questions from this chapter

Find the function value. Round to four decimal places. $$\cos 205.5^{\circ}$$

Use a calculator in Exercises \(81-84,\) but do not use the trigonometric function keys. Given that $$\begin{array}{l}\sin 35^{\circ}=0.5736 \\\\\cos 35^{\circ}=0.8192 \\\\\tan 35^{\circ}=0.7002\end{array}$$ find the trigonometric function values for \(215^{\circ}\).

Given the function value and the quadrant restriction, find \(\theta\). FUNCTION VALUE = \(\sec \theta=-1.0485\) INTERVAL = \(\left(90^{\circ}, 180^{\circ}\right)\) \(\boldsymbol{\theta}\) = ____

Find the signs of the six trigonometric function values for the given angles. $$-215^{\circ}$$

To find the distance between two points on the earth when their latitude and longitude are known, we can use a right triangle for an excellent approximation if the points are not too far apart. Point \(A\) is at latitude \(38^{\circ} 27^{\prime} 30^{\prime \prime} \mathrm{N},\) longitude \(82^{\circ} 57^{\prime} 15^{\prime \prime} \mathrm{W},\) and point \(B\) is at latitude \(38^{\circ} 28^{\prime} 45^{\prime \prime} \mathrm{N},\) longitude \(82^{\circ} 56^{\prime} 30^{\prime \prime} \mathrm{W}\). Find the distance from \(A\) to \(B\) in nautical miles. (One minute of latitude is one nautical mile.)
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