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Magazine Chapter 1: Problem 46
$$\text { Evaluate } \sin \frac{5 \pi}{12}$$
Short Answer
Expert verified
\( \sin \frac{5\pi}{12} = \frac{\sqrt{6} + \sqrt{2}}{4} \)
Step by step solution
01
Recognize the Angle
The goal is to evaluate \( \sin \frac{5\pi}{12} \). Let's recognize that this angle can be expressed in terms of known angles, like \( \frac{\pi}{4} \) (45 degrees) and \( \frac{\pi}{6} \) (30 degrees).
02
Express the Angle
Express \( \frac{5\pi}{12} \) as a sum of these known angles: \( \frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6} \). This follows since \( \frac{3\pi}{12} + \frac{2\pi}{12} = \frac{5\pi}{12}\).
03
Apply Sum of Angles Formula
Use the sum of angles formula for sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), where \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{6} \).
04
Substitute Known Values
Using the known values \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), \( \sin \frac{\pi}{6} = \frac{1}{2} \), and \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), substitute into the formula: \[ \sin \frac{5\pi}{12} = \sin \frac{\pi}{4} \cos \frac{\pi}{6} + \cos \frac{\pi}{4} \sin \frac{\pi}{6} \]
05
Simplify the Expression
Substituting in the known values, we have:\[ \sin \frac{5\pi}{12} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \]Simplify this to:\[ \sin \frac{5\pi}{12} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \]
06
Combine Terms
Combine the fractions:\[ \sin \frac{5\pi}{12} = \frac{\sqrt{6} + \sqrt{2}}{4} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum of Angles
In trigonometry, the sum of angles formula is a useful tool for breaking down complex angle measurements into simpler, more familiar ones. Specifically for the sine function, this formula is written as: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).This formula allows us to express the sine of a sum of two angles in terms of the sines and cosines of the individual angles \(a\) and \(b\). For this exercise, where we needed to evaluate \(\sin \frac{5\pi}{12}\), recognizing the angle as a sum of \(\frac{\pi}{4}\) (45 degrees) and \(\frac{\pi}{6}\) (30 degrees) was the key step that enabled the application of the sum of angles formula.Here are some highlights to remember:
- The sum of angles formula simplifies evaluation by relating unknown angles to known values.
- Breaking down the angle ensures simpler computation and better understanding of trigonometric properties.
- This approach can be applied to other trigonometric functions like cosine and tangent with their respective formulas.
Sine Function
The sine function is fundamental to trigonometry and is defined in terms of opposite over hypotenuse in a right triangle. Its periodic nature makes it useful in a variety of applications from physics to engineering.When working with specific values of sine, especially in radians, it's crucial to know the sine of common angles such as \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3},\) and \(\frac{\pi}{2}\). In this exercise, the known values \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\sin \frac{\pi}{6} = \frac{1}{2}\) were used effectively to compute the overall sine value of a more complex angle expression.Important points about the sine function:
- It's an odd function, so \(\sin(-\theta) = -\sin(\theta)\).
- Sine values are bounded between -1 and 1.
- Understanding sine at key angles makes solving trigonometric problems quicker and easier.
Angle Conversion
Angle conversion between degrees and radians is crucial in trigonometry since different problems might present angles in either unit. Understanding how to convert between the two enables consistent calculations and accuracy in results.The conversion factor is based on the relationship \(\pi\) radians = 180 degrees. Therefore, to convert from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\). Conversely, to convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).In practice:
- Knowing \(\frac{\pi}{6}\) is equivalent to 30 degrees and \(\frac{\pi}{4}\) is 45 degrees, makes following trigonometric problems involving these angles more intuitive.
- This exercise demonstrated converting angles to simpler forms (as sums) to solve for \(\sin \frac{5\pi}{12}\) using familiar angles and their trigonometric values.
- Make sure to always check your calculator's mode (degree vs radian) when performing trigonometric calculations.
