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Chapter 0: Problem 28

Find the exact values. Hint: Half-angle identities may be helpful. $$ \sin ^{2} \frac{\pi}{6} $$

Short Answer

Expert verified
\(\sin^2 \frac{\pi}{6} = \frac{1}{4}\)

Step by step solution

01

Recall the Half-Angle Identity

The half-angle identity for sine is given by: \[\sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}\] This identity will help us find \(\sin^2 \frac{\pi}{6}\).
02

Identify \(\theta\)

Since we are given \(\sin^2 \frac{\pi}{6}\), we can see that \(\frac{\pi}{6}\) is half of \(\frac{\pi}{3}\) because:\[2 \times \frac{\pi}{6} = \frac{\pi}{3}\]Thus, \(\theta = \frac{\pi}{3}\).
03

Find \(\cos\theta\)

We know from trigonometric values:\[\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\] Now, we'll use this in the half-angle identity.
04

Substitute into the Half-Angle Identity

Substitute \(\cos\theta = \frac{1}{2}\) into the half-angle identity:\[\sin^2\left(\frac{\pi}{6}\right) = \frac{1 - \frac{1}{2}}{2} = \frac{1/2}{2} = \frac{1}{4}\] This gives us the exact value of \(\sin^2 \frac{\pi}{6}\).

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Key Concepts

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

Half-Angle Identities
Half-angle identities are a key tool in trigonometry that help us calculate the values of trigonometric functions at half angles. These identities are particularly useful when the angle in question is not a special angle that you immediately recognize.

For example, the half-angle identity for sine is expressed as \( \sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2} \). This formula facilitates the calculation of the sine of half an angle using the cosine of the full angle.
  • This identity can be especially handy for angles such as \(\frac{\pi}{6}\), which are not directly available in the commonly used trigonometric tables.
  • By systematically applying half-angle identities, you can deduce precise trigonometric values without relying on calculators.
  • It's essential to remember that these identities derive from the angle addition and double angle formulas in trigonometry, reflecting the deep interconnectedness of trigonometric functions.
Sine Function
The sine function is a fundamental component of trigonometry, describing the relationship between certain angles and the ratios of sides in right-angle triangles. Sine functions are periodic with a period of \(2\pi\), meaning they repeat their values in regular intervals. The typical range of sine values is between -1 and 1.

The sine of an angle \(\theta\) in a right triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse. Mathematically, it is expressed as \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\). For unit circles, where the hypotenuse is 1, this definition simplifies the understanding and computation of the sine values.
  • The sine function is crucial for modeling oscillations, such as sound waves and light waves, in physics.
  • Knowing how to transform and manipulate sine functions, particularly through identities like the half-angle identity, improves your ability to solve a variety of trigonometric problems.
  • Understanding and memorizing the sine values of common angles, like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \pi/3,\) and \(\pi/2\) will greatly enhance your problem-solving efficiency.
Trigonometric Values
Trigonometric values constitute the numerical expressions of angles within the context of the trigonometric functions: sine, cosine, tangent, and their reciprocals. Each angle has a specific value in terms of these functions, and knowing these values is crucial for solving trigonometric equations efficiently.
  • The trigonometric value of \(\cos(\frac{\pi}{3})\) is \(\frac{1}{2}\), which is used in the half-angle identity to find \(\sin^2\left(\frac{\pi}{6}\right)\).
  • Recognizing these core trigonometric values allows you to work quickly through calculations that would otherwise require lengthy derivations.
  • A solid grasp of these values aids in understanding the deeper properties and behaviors of trigonometric functions in both mathematical and applied contexts.
Knowing these basic trigonometric values, like those for angles \(0, 30, 45, 60,\) and \(90\) degrees (or their radian equivalents), and how to apply them is fundamental to mastering trigonometry.

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Most popular questions from this chapter

Which of the following functions are odd? Even? Neither even nor odd? (a) \(f(x)=\frac{3 x}{x^{2}+1}\) (b) \(g(x)=|\sin x|+\cos x\) (c) \(h(x)=x^{3}+\sin x\) (d) \(k(x)=\begin{array}{c}x^{2}+1 \\ |x|+x^{4}\end{array}\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{2}-4 x+3 y^{2}=-2\)

Derive the formula \(A=\frac{1}{2} r^{2} t\) for the area of a sector of a circle. Here \(r\) is the radius and \(t\) is the radian measure of the central angle (see Figure 17 ).

Sketch the graph of each equation. $$ x=y^{2}-3 $$

Among all lines perpendicular to \(4 x-y=2,\) find the equation of the one that, together with the positive \(x\) - and \(y\) -axes, forms a triangle of area 8 .
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