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Magazine Chapter 1: Problem 52
True or false? Do not use a calculator. $$ \cos (13 \pi / 12)=-\cos (\pi / 12) $$
Short Answer
Expert verified
True
Step by step solution
01
Understand the Cosine Function Symmetry
Recall that the cosine function is an even function, which means that \(\text{cos}(-x) = \text{cos}(x)\). However, we need to understand how to deal with shifts in the angle.
02
Analyze the Argument Difference
The given angles are \(13 \pi / 12\) and \(\pi / 12\). Notice that \(13 \pi / 12 > \pi\), meaning it is in the third quadrant where cosine is negative.
03
Use Angle Addition Formulas
Recall the cosine addition formula: \(\text{cos}(a + b) = \text{cos}(a) \text{cos}(b) - \text{sin}(a) \text{sin}(b)\). Let's split the angle \(13 \pi / 12\) into \(\text{cos}(\pi + \pi / 12)\).
04
Simplify the Angle
Using \(a = \pi\) and \(b = \pi / 12\), apply \(\text{cos}(\pi + x) = -\text{cos}(x) \). Hence, \( \text{cos}(13 \pi / 12) = \text{cos}(\pi + \pi / 12) = -\text{cos}(\pi / 12) \).
05
Compare Both Sides
We have \( \text{cos}(13 \pi / 12) = -\text{cos}(\pi / 12)\) on both sides of the equation. This confirms the equality.
06
Conclusion
Since both sides of the equation are identical, the statement is true.
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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 \text{cos} is one of the basic trigonometric functions. It's crucial for understanding how angles and sides relate in various geometric shapes, such as triangles. The cosine function relates an angle in a right triangle to the ratio of the adjacent side to the hypotenuse. This is mathematically represented as: \[ \text{cos}(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \]
Cosine can be applied to any angle, not just those found in right triangles.
The function exhibits periodic behavior, repeating its values in a predictable pattern as the angle changes. This property forms part of what's known as the unit circle, which is instrumental in trigonometry.
Cosine can be applied to any angle, not just those found in right triangles.
The function exhibits periodic behavior, repeating its values in a predictable pattern as the angle changes. This property forms part of what's known as the unit circle, which is instrumental in trigonometry.
Even Function Property
A function is considered even if it satisfies the condition \( f(-x) = f(x) \). The cosine function is an example of this. This means that: \[ \text{cos}(-\theta) = \text{cos}(\theta) \]
This property is important because it shows that the cosine function is symmetric about the y-axis.
For instance, \(\text{cos}(-30^\text{掳}) = \text{cos}(30^\text{掳})\). This symmetry is valuable when solving problems involving trigonometric identities and transformations.
This property is important because it shows that the cosine function is symmetric about the y-axis.
For instance, \(\text{cos}(-30^\text{掳}) = \text{cos}(30^\text{掳})\). This symmetry is valuable when solving problems involving trigonometric identities and transformations.
Angle Addition Formula
The angle addition formula for cosine helps break complex angles into simpler parts. It states that: \[ \text{cos}(a + b) = \text{cos}(a) \text{cos}(b) - \text{sin}(a) \text{sin}(b) \]
This formula can be particularly useful in scenarios where angles are represented as sums or differences.
For example, in the problem, \(\text{cos}(13 \pi/12)\) can be expressed using \(\text{cos}(\pi + \pi/12)\). Applying the angle addition formula, we simplify to find: \[ \text{cos}(\pi + x) = -\text{cos}(x) \]
This resultant value simplifies many trigonometric expressions, making calculations easier.
This formula can be particularly useful in scenarios where angles are represented as sums or differences.
For example, in the problem, \(\text{cos}(13 \pi/12)\) can be expressed using \(\text{cos}(\pi + \pi/12)\). Applying the angle addition formula, we simplify to find: \[ \text{cos}(\pi + x) = -\text{cos}(x) \]
This resultant value simplifies many trigonometric expressions, making calculations easier.
Quadrants in Trigonometry
In trigonometry, the coordinate plane is divided into four quadrants, each with distinct characteristics affecting trigonometric functions. The four quadrants are:
Knowing which quadrant an angle lies in helps determine the sign of the trigonometric function.
For \(13 \pi/12\), since \(13 \pi/12\) is greater than \(\pi\), the angle falls in Quadrant III, where cosine is negative, confirming that: \[ \text{cos}(13 \pi/12) = -\text{cos}(\pi/12) \].
- **Quadrant I:** Both sine and cosine values are positive.
- **Quadrant II:** Sine is positive, but cosine is negative.
- **Quadrant III:** Both sine and cosine values are negative.
- **Quadrant IV:** Sine is negative, but cosine is positive.
Knowing which quadrant an angle lies in helps determine the sign of the trigonometric function.
For \(13 \pi/12\), since \(13 \pi/12\) is greater than \(\pi\), the angle falls in Quadrant III, where cosine is negative, confirming that: \[ \text{cos}(13 \pi/12) = -\text{cos}(\pi/12) \].
