/*! 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 65 Evaluate the following expressio... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 65

Evaluate the following expressions without using a calculator: (a) \(\sin \left(-\frac{5 \pi}{4}\right)\) (b) \(\cos \left(\frac{5 \pi}{6}\right)\) (c) \(\tan \left(\frac{\pi}{3}\right)\)

Short Answer

Expert verified
(a) \( \frac{\sqrt{2}}{2} \); (b) \( -\frac{\sqrt{3}}{2} \); (c) \( \sqrt{3} \).

Step by step solution

01

Identify Sine of Negative Angle

Recall that the sine function is odd, which means that \( \sin(-x) = -\sin(x) \). Thus, \( \sin \left(-\frac{5\pi}{4}\right) = -\sin \left(\frac{5\pi}{4}\right) \).
02

Locate Angle on the Unit Circle for Sine

Convert \( \frac{5\pi}{4} \) into degrees to understand its position on the unit circle. This is equivalent to 225掳. The reference angle for 225掳 is 45掳, which is in the third quadrant where sine is negative, meaning \( \sin \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).
03

Evaluate Sine Function

Using the property of the sine function being odd, \( \sin \left(-\frac{5\pi}{4}\right) = -(-\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{2} \).
04

Determine Cosine for Given Angle

Convert \( \frac{5\pi}{6} \) into degrees, which is 150掳. This falls in the second quadrant, where cosine is negative. The reference angle is 30掳 where \( \cos(30掳) = \frac{\sqrt{3}}{2} \). Thus, \( \cos(150掳) = -\frac{\sqrt{3}}{2} \).
05

Evaluate Cosine Expression

We then find \( \cos \left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \).
06

Identify Tangent Function

For \( \tan \left(\frac{\pi}{3}\right) \), convert \( \frac{\pi}{3} \) into degrees, which equals 60掳. The tangent of 60掳 is \( \sqrt{3} \).
07

Evaluate Tangent Expression

So, \( \tan \left(\frac{\pi}{3}\right) = \sqrt{3} \).

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Key Concepts

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

Unit Circle
The Unit Circle is an essential concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle is fundamental because it provides a way to obtain the sine, cosine, and tangent values for different angles:
  • The circle's circumference allows the angle to be measured in radians, which is more common in trigonometry than degrees.
  • Each point on the unit circle corresponds to \(x\) and \(y\) coordinates representing \( ext{cos}( heta)\) and \( ext{sin}( heta)\) respectively, for an angle \(\theta\).
  • The tangent is given by the ratio of sine to cosine, \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
The unit circle helps visualize where each trigonometric function is positive or negative across its four quadrants, providing a handy reference for solving trigonometric equations.
Sine Function
The Sine Function, denoted as \(\sin(x)\), is the vertical coordinate of a point on the unit circle.
  • This function is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.
  • It is an odd function which means \(\sin(-x) = -\sin(x)\).
  • The sine values vary between \(-1\) and \(1\), peaking at \(\pi/2\) and reaching a minimum at \(3\pi/2\).
For the original exercise, finding \(\sin\left(-\frac{5\pi}{4}\right)\), involves recognizing that \(\frac{5\pi}{4}\) lies in the third quadrant where sine is negative. Since sine is an odd function, \(\sin\left(-\frac{5\pi}{4}\right) = -(-\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{2}\).
Cosine Function
The Cosine Function, represented as \(\cos(x)\), describes the horizontal coordinate on the unit circle:
  • Similar to sine, the cosine function is periodic with a \(2\pi\) period.
  • Unlike sine, cosine is an even function, following \(\cos(-x) = \cos(x)\).
  • Values range from \(-1\) to \(1\), with \(1\) at \(0\) and \(2\pi\) radians, and \(-1\) at \(\pi\).
In the exercise, to find \(\cos\left(\frac{5\pi}{6}\right)\), you need to recognize that \(\frac{5\pi}{6}\) rests in the second quadrant, making cosine negative. With a reference angle of 30掳, and since \(\cos(30掳) = \frac{\sqrt{3}}{2}\), thus \(\cos(150掳) = -\frac{\sqrt{3}}{2}\).
Tangent Function
The Tangent Function, \(\tan(x)\), is the ratio of sine to cosine. This function is quite useful:
  • It's periodic, with a smaller cycle of \(\pi\) due to its repeating nature.
  • The function is undefined when cosine is zero, such as at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).
  • Tangent values can be any real number, unlike sine and cosine.
In the given problem, to evaluate \(\tan\left(\frac{\pi}{3}\right)\), knowing that \(\frac{\pi}{3}\) is 60掳 helps. With \(\tan(60掳) = \sqrt{3}\), you get \(\tan\left(\frac{\pi}{3}\right) = \sqrt{3}\).
Reference Angles
Reference angles are a key trigonometric concept helping simplify complex calculations. They allow us to refer back to key known angles in the first quadrant:
  • A reference angle is the acute angle formed by the terminal side of a given angle and the horizontal axis.
  • They are helpful because they exhibit the same trigonometric values (ignoring signs) as angles from other quadrants.
  • Signs differ based on which quadrant the original angle is located.
For example, if analyzing the angle like \(\frac{5\pi}{4}\), in the third quadrant, its reference angle is 45掳. Since the sine in the third quadrant is negative, this transforms \(\sin\left(\frac{5\pi}{4}\right)\) into \(-\frac{\sqrt{2}}{2}\). Reference angles help naturally convert angles to known trigonometric values, thus simplifying complex calculations.

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