Problem 35 - Contemporary Precalculus

Question

In Exercises \(31-36\), write the expression as a single real mum. ber. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\sin (3 \pi / 4) \cos (5 \pi / 6)-\cos (3 \pi / 4) \sin (5 \pi / 6)$$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\sin (3 \pi / 4) \cos (5 \pi / 6)-\cos (3 \pi / 4) \sin (5 \pi / 6)$$ without using decimal approximations. Answer: $$-\sin(\pi / 12)$$

Step 1: Identify the angle subtraction formula

We can see that the given expression is in the form of \(\sin A \cos B - \cos A \sin B\). So it matches the angle subtraction formula for sine, which is \(\sin(A - B) = \sin A \cos B - \cos A \sin B\). Let's identify the values of A and B. Given expression: $$\sin (3 \pi / 4) \cos (5 \pi / 6)-\cos (3 \pi / 4) \sin (5 \pi / 6)$$ Here, \(A = 3 \pi / 4\) and \(B = 5 \pi / 6\)

Step 2: Apply the angle subtraction formula

Now, we can apply the angle subtraction formula to find the sine of \((A - B)\): $$\sin(3 \pi / 4 - 5 \pi / 6)$$

Step 3: Calculate the angle difference

To find the difference of the angles, we need to find a common denominator for the two fractions. The common denominator is 12. So, $$3 \pi / 4 - 5 \pi / 6 = (3 \pi / 4) \cdot (3 / 3) - (5 \pi / 6) \cdot (2 / 2) = 9 \pi /12 - 10 \pi / 12$$ Now, calculate the difference: $$\frac{9 \pi}{12} - \frac{10 \pi}{12} = -\frac{\pi}{12}$$

Step 4: Find the sine of the angle difference

We found the angle difference to be \(-\pi / 12\). In the last step, we need to find the sine of this angle: $$\sin(-\pi / 12)$$ Since the sine function is odd, we can simplify this further: $$\sin(-\pi / 12) = -\sin(\pi / 12)$$ So the expression simplifies to a single real number: $$-\sin(\pi / 12)$$

Key Concepts

Angle Subtraction Formula

The angle subtraction formula is a key concept in trigonometry. It allows us to express the sine of a difference between two angles using the sines and cosines of the individual angles. Specifically, the formula is given by \( \sin(A - B) = \sin A \cos B - \cos A \sin B \).
This formula is incredibly useful because it simplifies many trigonometric problems, allowing us to work with basic functions rather than intricate combinations. To apply this formula effectively:

• Identify angles \( A \) and \( B \) from the problem you're working on.
• Use their sine and cosine to transform the expression into the subtraction of their angles.
• Calculate the difference and find the sine of the result.

In our exercise, this technique helps us convert the given complex expression into a more manageable form by determining \( \sin(3\pi/4 - 5\pi/6) \).
Understanding how to manipulate and apply this formula makes solving trigonometric equations much more straightforward, highlighting the elegance and efficiency of trigonometric identities.

Sine Function

The sine function is one of the fundamental functions in trigonometry. It is periodic and symmetric, usually defined for an angle \( \theta \) in a right triangle as the ratio of the length of the side opposite the angle to the hypotenuse. In the unit circle, it represents the y-coordinate of the corresponding point.

• The sine of 0 degrees is 0, while the sine of 90 degrees is 1.
• The function oscillates between -1 and 1.

When dealing with trigonometric identities, understanding the sine function's properties is crucial. It relates to the height of a point above the x-axis in a unit circle and has various applications such as calculating waveforms and rotations.
In the context of this problem, after applying the angle subtraction formula, we focused on finding \( \sin(-\pi/12) \). Recognizing symmetry and periodicity simplifies problems involving sine by reducing complex angles to smaller equivalents.
Knowing the sine function well offers a powerful toolset for solving numerous problems, both theoretical and applied.

Common Denominator

When working with fractions, particularly in trigonometry, finding a common denominator is an essential skill. In the case of angle subtraction, it's no different. A common denominator allows us to seamlessly subtract two fractional angles.
To illustrate, consider \( 3\pi/4 \) and \( 5\pi/6 \). The denominators 4 and 6 have a least common denominator of 12. By converting each fraction to this common denominator:

• Multiply \( 3\pi/4 \) by \( 3/3 \) to get \( 9\pi/12 \).
• Multiply \( 5\pi/6 \) by \( 2/2 \) to get \( 10\pi/12 \).

This conversion allows us to directly subtract to find the angle difference, leading to \(-\pi/12\). Finding a common denominator simplifies the arithmetic and is crucial for accuracy. Understanding this step is vital for handling more complex problems where angle manipulation is needed.

Odd Function

An odd function in mathematics, such as the sine function, has a distinctive symmetric property: \( f(-x) = -f(x) \). This means odd functions are symmetric around the origin. The sine function, \( \sin(-\theta) = -\sin(\theta) \), exhibits this behavior.
In our problem, recognizing the sine function as odd allows for simplification. Given \( \sin(-\pi/12) \), using its odd property gives us \( -\sin(\pi/12) \).

• This property helps in reducing calculations when negative angles are involved.
• It emphasizes symmetry, simplifying tasks in both graphical and algebraic perspectives.

An odd function's inherent symmetry is a powerful tool, especially in trigonometry. It allows us to handle negative angles gracefully, turning potential complexities into straightforward solutions. Understanding this function type is crucial in simplifying and solving trigonometric equations efficiently.