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Chapter 6: Problem 25

Evaluate the expression without using a calculator. $$\sin \frac{\pi}{6}+\cos \frac{\pi}{6}$$

Short Answer

Expert verified
\( \frac{1 + \sqrt{3}}{2} \)

Step by step solution

01

Recognize Standard Angles and Values

The angles \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \) correspond to the standard angles in trigonometry with values that are often memorized. \( \frac{\pi}{6} \) is equivalent to 30 degrees, and \( \frac{\pi}{3} \) is equivalent to 60 degrees.
02

Calculate \( \sin \frac{\pi}{6} \)

The sine of \( \frac{\pi}{6} \) or 30 degrees is a standard value and is known to be \( \frac{1}{2} \). Thus, \( \sin \frac{\pi}{6} = \frac{1}{2} \).
03

Calculate \( \cos \frac{\pi}{6} \)

Similarly, the cosine of \( \frac{\pi}{6} \) or 30 degrees is also a standard value known to be \( \frac{\sqrt{3}}{2} \). Therefore, \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \).
04

Add the Evaluated Values

Now, we add the results of \( \sin \frac{\pi}{6} \) and \( \cos \frac{\pi}{6} \). This gives us: \[\sin \frac{\pi}{6} + \cos \frac{\pi}{6} = \frac{1}{2} + \frac{\sqrt{3}}{2}\]Combine the fractions to get: \[\frac{1 + \sqrt{3}}{2}\].

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Key Concepts

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

Standard Angles
In trigonometry, standard angles or special angles are commonly used because their trigonometric values are simple and easy to remember. These angles often appear in both degree and radian measure, making them fundamental in solving problems without a calculator.
Standard angles include angles like 0掳, 30掳, 45掳, 60掳, and 90掳, as well as their counterparts in radians: 0, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \).
Learning to recognize these angles quickly can help you solve trigonometric equations faster. Here鈥檚 a tip: the smaller the angle, the smaller the sine value, while the cosine decreases as you move from 0 to 90 degrees.
Sine and Cosine Values
When evaluating trigonometric functions, certain angles yield standard sine and cosine values. Let's take a closer look at understanding these values without a calculator.
For the angle \( \frac{\pi}{6} \) or 30 degrees, the sine value is \( \frac{1}{2} \) and the cosine value is \( \frac{\sqrt{3}}{2} \). This is often remembered as part of trigonometric identities.
  • Sine values start at 0 and increase to 1 as the angle increases towards \( \frac{\pi}{2} \) or 90 degrees.
  • Conversely, cosine values start at 1 and decrease to 0 over the same range.
These values are derived from the unit circle, a helpful visualization tool in understanding trigonometric functions across different quadrants.
Angle Conversion
Degrees and radians are two different units used to measure angles. Being able to convert between them is useful for solving various trigonometric problems.
To convert degrees to radians, multiply by \( \frac{\pi}{180} \). Conversely, to convert radians to degrees, multiply by \( \frac{180}{\pi} \).
For example, when converting \( \frac{\pi}{6} \) radians to degrees:
\( \frac{\pi}{6} \times \frac{180}{\pi} = 30 \) degrees.
This conversion demonstrates why \( \frac{\pi}{6} \) is often associated with 30 degrees. Mastery of angle conversion allows easy transition between systems, aiding in clear comprehension of trigonometric identities and functions.

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Most popular questions from this chapter

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\tan \theta=-\frac{3}{2}, \quad \cos \theta>0$$

A triangle has an area of 16 in \(^{2}\), and two of the sides of the triangle have lengths 5 in. and 7 in. Find the angle included by these two sides.

Rewrite the expression as an algebraic expression in \(x\). $$\sin \left(\tan ^{-1} x\right)$$

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The range \(R\) and height \(H\) of a shot put thrown with an initial velocity of \(v_{0}\) ft's at an angle \(\theta\) are given by $$\begin{array}{l} R=\frac{v_{0}^{2} \sin (2 \theta)}{g} \\ H=\frac{v_{0}^{2} \sin ^{2} \theta}{2 g} \end{array}$$ On the earth \(q=32 \mathrm{ft} / \mathrm{s}^{2}\) and on the moon \(g=5.2 \mathrm{ft} / \mathrm{s}^{2} .\) Find the range and height of a shot put thrown under the given conditions. (a) On the earth with \(v_{0}=12 \mathrm{ft} / \mathrm{s}\) and \(\theta=\pi / 6\) (b) On the moon with \(u_{b}=12\) ft \(/ s\) and \(\theta=\pi / 6\)
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