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Chapter 1: Problem 17

\(\sin\) \(30^{\circ}\)

Short Answer

Expert verified
The sine of 30 degrees is 0.5.

Step by step solution

01

Recognise the standard angle

Here, the angle provided is 30 degrees. It is one of the standard angles for which we can directly know the values of sine. These standard angles include 0, 30, 45, 60, and 90 degrees.
02

Recall the value of sine

From the special right triangle definition or the unit circle definition of sine or memorizing the trigonometric table, we know that \(\sin(30^{\circ})=0.5\).

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Key Concepts

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

Standard Angles
Standard angles are specific angles that are frequently utilized in trigonometry due to their simple sine, cosine, and tangent values. These angles include \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\). Memorizing these angles simplifies solving trigonometric equations.
  • Commonly used in mathematical problems.
  • Often appear in unit circle and special triangles discussions.
By remembering the trigonometric values of these standard angles, you can quickly find solutions without complex calculations. For example, \(\sin 30^\circ\) is known to be \(0.5\) just by recognizing the angle.
Sine Function
The sine function is a fundamental part of trigonometry that relates a right-angled triangle's structure to angles. Given an angle \(\theta\), sine is defined as the ratio of the length of the opposite side to the hypotenuse.
  • Helps in determining height or distance in applications.
  • Key function in waveforms and oscillatory motion in physics.
In our exercise, knowing that \(\sin(30^\circ) = 0.5\) allows us to understand how much of the unit circle is traversed and makes it easy to visualize the triangle's proportions.
Unit Circle
The unit circle is a pivotal concept in trigonometry that provides a comprehensive means of understanding trigonometric functions over all angles, not just standard ones. A circle with a radius of 1 centered at the origin, it relates angles to coordinates.
  • Coordinates on the circle provide values for sine and cosine.
  • Every angle corresponds to a point on the circle.
For example, in the unit circle, \(\sin(30^\circ)\) corresponds to the \(y\)-coordinate of the point where a \(30^\circ\) angle intercepts the circle, giving a value of \(0.5\). This illustrates why \(30^\circ\) is such a straightforward angle to work with.
Special Right Triangles
Special right triangles are those with angles in standard measures, such as \(30^\circ-60^\circ-90^\circ\) and \(45^\circ-45^\circ-90^\circ\). They serve as foundational tools in solving trigonometric problems.
  • Help simplify understanding of trigonometric ratios.
  • Often used in conjunction with the unit circle.
In a \(30^\circ-60^\circ-90^\circ\) triangle, the side opposite the \(30^\circ\) angle is half the hypotenuse. This leads directly to knowing that \(\sin(30^\circ) = 0.5\). Recognizing these patterns aids in solving a wide range of geometric problems without extensive calculation.

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Most popular questions from this chapter

The height of a radio transmission tower is 70 meters, and it casts a shadow of length 30 meters (see figure). Find the angle of elevation of the sun.

In Exercises 89 and 90, write the function in terms of the sine function by using the identity $$ A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right) $$ Use a graphing utility to graph both forms of the function. What does the graph imply? $$ f(t)=4 \cos \pi t+3 \sin \pi t $$

In Exercises 109-112, sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side. Then find the other five trigonometric functions of \(\boldsymbol{\theta}\). $$ \sin \theta=\frac{3}{4} $$

In Exercises 19-34, use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arcsin (-0.75) $$

$$ \text { In Exercises 77-82, sketch a graph of the function. } $$ $$ f(x)=\arccos \frac{x}{4} $$
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