/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 What is \(\sin \frac{\pi}{12}\) ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 60

What is \(\sin \frac{\pi}{12}\) given that \(\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}\) and that \(\sin (\alpha-\beta)=(\sin \alpha)(\cos \beta)-(\cos \alpha)(\sin \beta) ?\) (Note: You may use the following table of values.) table can't copy F. \(\frac{1}{4}\) G. \(\frac{1}{2}\) H. \(\frac{\sqrt{3}-2}{4}\) J. \(\frac{\sqrt{3}-\sqrt{2}}{2}\) K. \(\frac{\sqrt{6}-\sqrt{2}}{4}\)

Short Answer

Expert verified
Answer: \(\sin(\frac{\pi}{12}) = \frac{\sqrt{6}-\sqrt{2}}{4}\)

Step by step solution

01

Recall the sum/difference formula for sine

The given sum/difference formula for sine is $$\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)$$ We'll use this formula to find the sine of \(\frac{\pi}{12}\).
02

Rewrite \(\frac{\pi}{12}\) as the difference of two known angles

We are given that $$\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$$ so we have \(\alpha = \frac{\pi}{3}\) and \(\beta = \frac{\pi}{4}\).
03

Find the sine and cosine values for \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\)

Use the given table to find the sine and cosine values for these two angles: $$\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \quad \cos \frac{\pi}{3} = \frac{1}{2}$$ $$\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \quad \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$$
04

Substitute the values into the sum/difference formula

Substitute the values of \(\alpha\) and \(\beta\) as well as their sine and cosine values into the formula: $$\sin(\frac{\pi}{12}) = \sin(\frac{\pi}{3} - \frac{\pi}{4}) = \sin(\frac{\pi}{3})\cos(\frac{\pi}{4}) - \cos(\frac{\pi}{3})\sin(\frac{\pi}{4})$$ Now, plug in the sine and cosine values: $$\sin(\frac{\pi}{12}) = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)$$
05

Simplify the expression

Simplify the expression by multiplying the fractions and combining the terms: $$\sin(\frac{\pi}{12}) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$$ Put everything under the same denominator: $$\sin(\frac{\pi}{12}) = \frac{\sqrt{6}-\sqrt{2}}{4}$$ By comparing it to the given options, we can see this value matches option K. So, the value of \(\sin(\frac{\pi}{12})\) is \(\boxed{\text{K. }\frac{\sqrt{6}-\sqrt{2}}{4}}\).

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Key Concepts

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

Understanding Trigonometry
Trigonometry is a branch of mathematics that delves into the relationships between angles and sides of triangles, primarily right-angled triangles. It encompasses a wide range of functions such as sine, cosine, and tangent, which are essential to various fields including physics, engineering, and even art and music.

One of the fundamental concepts is that trigonometric functions are cyclical and can be represented on the unit circle, which has a radius of one. Angles in trigonometry are typically measured in radians, where the complete cycle around a circle corresponds to \(2\text{蟺}\) radians. Understanding the unit circle is critical for mastering trigonometry, as it provides visual insight into how the sine and cosine functions behave as angles increase or decrease. This understanding is key when you鈥檙e tackling problems on the ACT Math section, which often requires you to calculate angles and side lengths by utilizing these trigonometric functions.
Mastering Sum and Difference Formulas
Sum and difference formulas are trigonometric identities that allow us to find the sine, cosine, or tangent of the sum or difference of two angles. A common use of these formulas is to simplify the calculation of trigonometric functions for angles not found on the standard unit circle values.

For instance, the sine of the sum or difference of two angles \(伪\) and \(尾\) can be written as \(\sin(伪 卤 尾) = \sin伪\cos尾 卤 \cos伪\sin尾\). These formulas facilitate the process of breaking down unfamiliar angles into known components. On the ACT Math, this can become especially useful when you encounter complex angles and are expected to calculate without a calculator. By practicing these formulas, you ensure that you can efficiently work through this section of the exam without losing precious time.
Sine Function Calculations
The sine function is one of the primary trigonometric functions, defined for a given angle as the ratio of the length of the opposite side to the hypotenuse in a right triangle. However, for ACT Math preparation, it is essential to be familiar with the sine values for common angles such as \(30掳\), \(45掳\), and \(60掳\) (or \(\frac{\text{蟺}}{6}\), \(\frac{\text{蟺}}{4}\), and \(\frac{\text{蟺}}{3}\) in radians), as well as understanding how to apply the sum and difference formulas.

To calculate sine values efficiently, one must not only memorize the unit circle and sine value table but also understand how to use this information alongside trigonometric identities. This statistical approach to learning trigonometry can significantly improve performance, especially in a high-pressure test environment, allowing for quick and accurate computation of complex sine functions.

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