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Magazine Chapter 6: Problem 8
Find the exact values. (a) \(\cos 225^{\circ}-\cos 30^{\circ}\) (b) \(\left.\cos 195^{\circ} \text { (use } 195^{\circ}=225^{\circ}-30^{\circ}\right)\)
Short Answer
Expert verified
(a) \(-\frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2}\), (b) \(-\frac{\sqrt{6} + \sqrt{2}}{4}\)
Step by step solution
01
Understanding the Problem (a)
To solve \( \cos 225^{\circ} - \cos 30^{\circ} \), we will first identify the exact trigonometric values for these angles. These values are found using the unit circle.
02
Calculate \( \cos 225^{\circ} \)
The angle \( 225^{\circ} \) is in the third quadrant, where cosine is negative. It is equivalent to \( 180^{\circ} + 45^{\circ} \), resulting in \( \cos 225^{\circ} = -\frac{\sqrt{2}}{2} \).
03
Calculate \( \cos 30^{\circ} \)
The angle \( 30^{\circ} \) is in the first quadrant, where cosine is positive. The exact value is \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).
04
Solving \( \cos 225^{\circ} - \cos 30^{\circ} \)
Subtract the value of \( \cos 30^{\circ} \) from \( \cos 225^{\circ} \). So, \(-\frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} = -\frac{\sqrt{2} - \sqrt{3}}{2}\).
05
Strategy for Problem (b)
We will use the cosine angle subtraction identity to find \( \cos 195^{\circ} \) because \( 195^{\circ} = 225^{\circ} - 30^{\circ} \). The identity is \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
06
Calculate \( \cos 195^{\circ} \) Using Identity
Using \( \cos 225^{\circ} = -\frac{\sqrt{2}}{2} \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), \( \sin 225^{\circ} = -\frac{\sqrt{2}}{2} \), and \( \sin 30^{\circ} = \frac{1}{2} \), substitute into the formula: \( \cos 195^{\circ} = \left(-\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(-\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = -\frac{\sqrt{6} + \sqrt{2}}{4} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cosine Function
The cosine function, denoted as \( \cos \theta \), is a vital component in trigonometry. It relates the angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse in a right-angled triangle. The cosine function is periodic with a period of \( 360^\circ \text{ or } 2\pi \), meaning it repeats its values over these intervals.
- In the unit circle, the cosine value represents the \( x \)-coordinate of the point at a given angle \( \theta \) from the positive \( x \)-axis.
- The cosine function is even, which implies \( \cos(-\theta) = \cos(\theta) \), helping in simplifying expressions.
- Understanding cosine's behavior in different quadrants is crucial:
- First Quadrant (\(0^{\circ} \text{ to } 90^{\circ}\)): Cosine values are positive.
- Second Quadrant (\(90^{\circ} \text{ to } 180^{\circ}\)): Cosine values are negative.
- Third Quadrant (\(180^{\circ} \text{ to } 270^{\circ}\)): Cosine values remain negative.
- Fourth Quadrant (\(270^{\circ} \text{ to } 360^{\circ}\)): Cosine values return to positive.
Unit Circle
The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane \((0, 0)\). This simple yet powerful tool allows us to define trigonometric functions for all angles through circular motion.
- Each point \((x, y)\) on the unit circle corresponds to the cosine and sine values, where \( x = \cos \theta \) and \( y = \sin \theta \).
- The unit circle helps us understand angle measures:
- Angles are measured counterclockwise from the positive \( x \)-axis.
- Coordinates for well-known angles are easily memorized: for example, \( (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \) for \( 45^{\circ} \) and \( (\frac{1}{2}, \frac{\sqrt{3}}{2}) \) for \( 30^{\circ} \).
- Using the unit circle, trigonometric functions can be evaluated for any angle, thus solving for both standard and non-standard angles.
Angle Subtraction Formula
The angle subtraction formula is a significant tool in trigonometry. This formula provides the exact value of trigonometric functions for angles expressed as the difference of two known angles. Specifically, for cosine, the formula is:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
- By applying this identity, one can simplify and calculate the cosine of complex angles without direct measurement.
- For instance, if we need \( \cos 195^{\circ} \) knowing it can be expressed as \( 225^{\circ} - 30^{\circ} \), use:
- \(\cos 225^{\circ} = -\frac{\sqrt{2}}{2}\)
- \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
- \(\sin 225^{\circ} = -\frac{\sqrt{2}}{2}\)
- \(\sin 30^{\circ} = \frac{1}{2}\)
- Substitute these into the formula, attaining: \[ \cos 195^{\circ} = -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = -\frac{\sqrt{6} + \sqrt{2}}{4} \]
