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Chapter 8: Problem 26

Find the exact value of each expression. $$ \sin 20^{\circ} \cos 80^{\circ}-\cos 20^{\circ} \sin 80^{\circ} $$

Short Answer

Expert verified
-\(\frac{\text{\sqrt{3}}}{2}\)

Step by step solution

01

Identify the Trigonometric Identity

The expression \(\text{sin}\theta \text{cos}\beta - \text{cos}\theta \text{sin}\beta\) is a known trigonometric identity. It corresponds to the angle subtraction formula for sine: \(\text{sin}(\theta - \beta)\).
02

Apply the Identity

Rewrite the given expression using the identity identified in Step 1. Here, \(\theta = 20^{\text{\circ}}\) and \(\beta = 80^{\text{\circ}}\). Apply the identity to get: \(\text{sin}(20^{\text{\circ}} - 80^{\text{\circ}})\).
03

Subtract Angles

Subtract the angles in the sine function: \(20^{\text{\circ}} - 80^{\text{\circ}} = -60^{\text{\circ}}\). Thus, the expression simplifies to \(\text{sin}(-60^{\text{\circ}})\).
04

Evaluate the Sine Function

Recall that the sine function is odd, so \(\text{sin}(-\theta) = -\text{sin}(\theta)\). Therefore, \(\text{sin}(-60^{\text{\circ}}) = -\text{sin}(60^{\text{\circ}})\).
05

Find the Value of \(\text{sin}(60^{\text{\circ}})\)

Recall that \(\text{sin}(60^{\text{\circ}}) = \frac{\text{\sqrt{3}}}{2}\). Thus, \(-\text{sin}(60^{\text{\circ}}) = -\frac{\text{\sqrt{3}}}{2}\).
06

Combine Results

Combine the result from Step 5 to get the final value of the expression: \(-\frac{\text{\sqrt{3}}}{2}\).

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Key Concepts

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

Angle Subtraction Formula
In trigonometry, the angle subtraction formula is a pivotal identity. It simplifies the expression of the sine function when dealing with the difference of two angles. The formula is given by: \(\sin(\theta - \beta) = \sin\theta \cos\beta - \cos\theta \sin\beta\). This identity enables the simplification of trigonometric expressions and calculations by converting subtraction of angles into a product of sines and cosines. For example, in the exercise provided, \(\sin(20^{\circ} \cos 80^{\circ}-\cos 20^{\circ} \sin 80^{\circ})\) can be rewritten using this formula as \(\sin(20^{\circ} - 80^{\circ})\).
Sine Function Properties
Understanding the properties of the sine function is crucial for evaluating and transforming trigonometric expressions. The sine function has several key properties:
  • It is periodic with a period of \(2\pi\).
  • It is an odd function, meaning \(\sin(-\theta) = -\sin(\theta)\).
  • Its values range between -1 and 1, inclusive.
In the context of the problem, we use the property that the sine function is odd. This helps in simplifying \(\sin(-60^{\circ})\) as: \(\sin(-60^{\circ}) = -\sin(60^{\circ})\). Recognizing and applying these properties properly can make solving trigonometric problems much easier.
Evaluating Trigonometric Functions
Evaluating trigonometric functions often involves calculating the values of trigonometric ratios for specific angles. For instance, some well-known values for sine are:
  • \sin(30^{\circ})=\frac{1}{2}
  • \sin(45^{\circ})=\frac{\sqrt{2}}{2}
  • \sin(60^{\circ})=\frac{\sqrt{3}}{2}
In the problem we are analyzing, the goal is to find the exact value of \(\sin(-60^{\circ})\). Recognizing that \(\sin(60^{\circ})=\frac{\sqrt{3}}{2}\), and using the property that \(\sin(-\theta) = -\sin(\theta)\), we find: \(\sin(-60^{\circ}) = -\frac{\sqrt{3}}{2}\). This method illustrates the importance of knowing basic trigonometric function values and their properties to evaluate expressions efficiently.

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