Problem 36 - Contemporary Precalculus

Question

In Exercises \(31-36\), write the expression as a single real number. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\sin (-7 \pi / 3) \cos (5 \pi / 4)+\cos (-7 \pi / 3) \sin (5 \pi / 4)$$

Step-by-Step Solution

Answer

Answer: The value of the trigonometric expression is $$\frac{1}{2}\left(\sqrt{6} - \sqrt{2}\right)$$

Step 1: Identify A and B values for the angle addition formula

To rewrite the given expression using the angle addition formula, we will set: $$A = -\frac{7\pi}{3}$$ $$B = \frac{5\pi}{4}$$

Step 2: Apply the angle addition formula

Using the angle addition formula for sine, the expression can be written as: $$\sin \left(-\frac{7\pi}{3} + \frac{5\pi}{4}\right)$$

Step 3: Find the sum of angles A and B

Now we need to add the angles A and B: $$-\frac{7\pi}{3} + \frac{5\pi}{4} = \frac{-28\pi + 15\pi}{12} = -\frac{13\pi}{12}$$

Step 4: Calculate the sine value of the angle

Now we need to find the sine value of the new angle -\(\frac{13\pi}{12}\). Since the sine function has a period of \(2\pi\), we can add \(2\pi\) to the angle without changing the sine value: $$\sin \left(-\frac{13\pi}{12} \right) = \sin \left(-\frac{13\pi}{12} + 2\pi \right) = \sin \frac{\pi}{12}$$ Now, we know the value of sine of \(\frac{\pi}{12}\): $$\sin \frac{\pi}{12} = \frac{1}{2}\left(\sqrt{6} - \sqrt{2}\right)$$

Step 5: Write the final answer

The expression $$\sin (-7 \pi / 3) \cos (5 \pi / 4)+\cos (-7 \pi / 3) \sin (5 \pi / 4)$$ simplifies to a single real number as: $$\frac{1}{2}\left(\sqrt{6} - \sqrt{2}\right)$$

Key Concepts

Angle Addition Formula

The angle addition formula is a fundamental tool in trigonometry that helps us to combine two angles. It's most commonly used for sine and cosine functions. For sine, the formula is given by: \ \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). This formula is particularly useful when you encounter trigonometric expressions that involve the sum or difference of two angles.

This identity allows us to simplify expressions and solve equations by reducing complex trigonometric expressions into more manageable terms. In the original exercise, we utilized this formula to transform the expression \( \sin(-\frac{7\pi}{3}) \cos(\frac{5\pi}{4}) + \cos(-\frac{7\pi}{3}) \sin(\frac{5\pi}{4}) \) into a single sine function of the sum of angles \( -\frac{7\pi}{3} \) and \( \frac{5\pi}{4} \). This step was crucial in reaching the final simplified number. By understanding and applying the angle addition formula, complex problems become more accessible and easier to tackle.

Sine Function

The sine function is one of the basic trigonometric functions, often abbreviated as "sin". It's integral in understanding the properties of waves, circles, and oscillatory motions. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. For the unit circle, this becomes the y-coordinate of a point on the circle.

The sine function is periodic, meaning it repeats its values in regular intervals. One complete cycle occurs over an interval of \( 2\pi \) radians. This property is helpful when dealing with angles outside the primary range \( [0, 2\pi] \), enabling us to find equivalent angles with the same sine value. In our exercise, we applied this property to calculate the sine of \(-\frac{13\pi}{12}\). By adding \(2\pi\), we effectively shifted the angle into a familiar range without altering its sine value, arriving at \( \sin \frac{\pi}{12} \).

• The sine function alternates between -1 and 1.
• The graph of the sine function crosses the x-axis at integer multiples of \( \pi \).
• Key values occur at \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), and \( 2\pi \).

Periodicity of Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are known for their periodic nature. This means they repeat their values after a certain interval, known as the "period". For sine and cosine functions, this period is \(2\pi\), while for tangent, it is \(\pi\).

This periodicity is incredibly useful in solving trigonometric problems because it allows us to handle infinite angle domains in a finite manner. In practical terms, it means that for any angle \( \theta \), \( \sin(\theta + 2\pi n) = \sin(\theta) \), where \( n \) is an integer. Consequently, angles can be co-terminal, sharing properties and measurements of interest, which helps streamline the problem-solving process.

For example, in our exercise, we recalculated \( \sin(-\frac{13\pi}{12}) \) by utilizing the periodicity of the sine function, thereby simplifying it to \( \sin \frac{\pi}{12} \), which is easier to evaluate directly from known values.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is an essential skill in trigonometry because it makes complex equations easier to manage and understand. This process often involves rewriting expressions using identities, reducing angles within specific ranges, or breaking down compound angles using formulas like the angle addition formula.

In our exercise, the expression \( \sin(-\frac{7\pi}{3}) \cos(\frac{5\pi}{4}) + \cos(-\frac{7\pi}{3}) \sin(\frac{5\pi}{4}) \) initially seemed complex. However, by applying the angle addition formula and utilizing trigonometric properties such as periodicity and known sine values, we transformed it into a much simpler single real number: \( \frac{1}{2}(\sqrt{6} - \sqrt{2}) \).

• Identify and use relevant trigonometric identities.
• Simplify by reducing angles using periodic properties.
• Aim to combine terms into familiar expressions or simpler angles.

By mastering these techniques, you can tackle a wide range of trigonometric problems with confidence and precision.