Problem 12 - Calculus with Concepts in Calculus

Question

Approximate the number. $$ \sin \frac{9 \pi}{20} \cos \frac{9 \pi}{30} $$

Step-by-Step Solution

Answer

0.5805

Step 1: Convert angles to degrees

Convert the given angles from radians to degrees. Note that \[ \frac{9 \pi}{20} \text{ radians} = \frac{9 \times 180}{20} \text{ degrees} = 81^{\circ} \]and \[ \frac{9 \pi}{30} \text{ radians} = \frac{9 \times 180}{30} \text{ degrees} = 54^{\circ}. \]

Step 2: Use trigonometric identities

We will use the identity: \[ \sin A \cos B = \frac{1}{2} \left[ \sin(A+B) + \sin(A-B) \right] \]Substitute \(A = 81^{\circ}\) and \(B = 54^{\circ}\), so we have:\[ \sin 81^{\circ} \cos 54^{\circ} = \frac{1}{2} \left[ \sin(81^{\circ} + 54^{\circ}) + \sin(81^{\circ} - 54^{\circ}) \right]. \]

Step 3: Calculate the angles

Calculate \(81^{\circ} + 54^{\circ} = 135^{\circ}\) and \(81^{\circ} - 54^{\circ} = 27^{\circ}\).

Step 4: Evaluate the sine of the angles

Use a calculator or trigonometric tables to find \(\sin 135^{\circ}\) and \(\sin 27^{\circ}\). We know:\[ \sin 135^{\circ} = \sin (180^{\circ} - 45^{\circ}) = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \]and typically from a calculator,\[ \sin 27^{\circ} \approx 0.454. \]

Step 5: Simplify and calculate the expression

Substitute the values found into the expression from Step 2:\[ \sin 81^{\circ} \cos 54^{\circ} = \frac{1}{2} \left[ \frac{\sqrt{2}}{2} + 0.454 \right]. \]Calculate the result:\[ \frac{1}{2} \left[ \frac{\sqrt{2}}{2} + 0.454 \right] \approx \frac{1}{2} \left[ 0.707 + 0.454 \right] = \frac{1}{2} \times 1.161 = 0.5805. \]

Key Concepts

Angle Conversion

When working with trigonometric functions, it's essential to know how to convert angles between radians and degrees. This conversion helps in understanding and comparing results more intuitively. The conversion formula between radians and degrees is derived from the fact that a full circle is 360 degrees or \(2\pi\) radians. The formula is:

• Degrees to Radians: \(\text{{radians}} = \text{{degrees}} \times \frac{\pi}{180}\)
• Radians to Degrees: \(\text{{degrees}} = \text{{radians}} \times \frac{180}{\pi}\)

In our exercise, the angles \(\frac{9\pi}{20}\) and \(\frac{9\pi}{30}\) radians need to be converted to degrees to facilitate further calculation. Using the radians to degrees conversion formula, we calculated them as 81° and 54° respectively. This step lays the groundwork for applying trigonometric identities effectively.
Mastering angle conversion is crucial for solving problems involving periodic functions and understanding their behavior in different units.

Sine and Cosine Functions

The sine and cosine functions are among the fundamental building blocks in trigonometry. These functions describe the ratio of sides of a right triangle with respect to its angles.

• \(\sin(\theta)\) gives the ratio of the opposite side to the hypotenuse.
• \(\cos(\theta)\) gives the ratio of the adjacent side to the hypotenuse.

In trigonometry, several identities involving sine and cosine simplify complex expressions. One such identity is:\[ \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \] This identity is particularly useful for simplifying products of sines and cosines of different angles, as in our exercise. By substituting the angles (81° and 54°) into this identity, we express \(\sin 81^{\circ} \cos 54^{\circ}\) as a simpler sum of sines. This step is crucial before evaluating individual sine values, as it transforms a multiplication problem into an addition problem, often making it easier to calculate or approximate.

Approximation Techniques

Sometimes, trigonometric values cannot be expressed exactly in terms of radicals or simple fractions. In such cases, approximation techniques become valuable. For example, common angles like 30°, 45°, and 60° have known sine and cosine values. However, angles like 27° often require approximation using a calculator. To tackle the approximation:

• Identify angles that are standard or can be broken down into known values (like \(135^{\circ} = 180^{\circ} - 45^{\circ}\)).
• Use trigonometric identities, such as supplementary and complementary angle identities, to simplify the calculation.
• When necessary, use a calculator to find approximate trigonometric values with reasonable precision. In our case, \(\sin 27^{\circ} \approx 0.454\).

Once approximate values are obtained, they can be plugged back into expressions derived from identities to compute an overall approximation. This results in a final number that closely represents the solution to the original problem. Understanding and applying approximation techniques is crucial for working efficiently with more complex or non-standard angle measurements.