Problem 12 - Precalculus with Limits

Question

In Exercises 7 - 12, find the exact value of each expression. (a) \( \sin\left(315^\circ - 60^\circ\right) \) (b) \( \sin 315^\circ - 60^\circ \)

Step-by-Step Solution

Answer

The exact value of \( \sin\left(315^\circ - 60^\circ\right) \) is -\( \sqrt{2} / 2 \) and the exact value of \( \sin 315^\circ - 60^\circ \) is \( - \sqrt{2} / 2 - 60 \).

Step 1: Understand the Difference in Syntax

The first expression \( \sin\left(315^\circ - 60^\circ\right) \) means that the subtraction occurs within the sin function. On the other hand, \( \sin 315^\circ - 60^\circ \) directs that the subtraction is done after calculating the sin of 315 degrees.

Step 2: Solve for \( \sin\left(315^\circ - 60^\circ\right) \)

Subtract 60 degrees from 315 degrees within the sin function. This results in \( \sin 255^\circ \). Using the unit circle or a trigonometric table, we know that \( \sin 255^\circ = - \sqrt{2} / 2 \).

Step 3: Solve for \( \sin 315^\circ - 60^\circ \)

First calculate the sin of 315 degrees. Using the unit circle or a trigonometric table, we know that \( \sin 315^\circ = - \sqrt{2} / 2 \). Then subtract 60 from this value to get \( - \sqrt{2} / 2 - 60 \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing all the angles found on a circle with a radius of one, centered at the origin of a Cartesian coordinate system. The circle aids in defining sine, cosine, and tangent for all angles.
On the unit circle, each angle corresponds to a point whose x-coordinate represents the cosine of that angle, and the y-coordinate represents the sine. For example, the angle 0° is at the point (1, 0), while 90° is at the point (0, 1). This makes understanding trigonometric functions at standard angles easier.

• Angles are typically measured in degrees or radians.
• The positions of key angles such as 30°, 45°, 60°, and their corresponding coordinates are often memorized to simplify solving problems.

The unit circle also simplifies the process of finding the exact trigonometric values of angles, particularly when dealing with angle sums or differences.

Subtraction Within Trigonometric Functions

Subtraction within trigonometric functions, such as in \( \sin(315^\circ - 60^\circ) \), involves performing the arithmetic operation before applying the trigonometric function. This subtraction is an essential part of the trigonometric identity application.
In this process, you first determine the angle created by subtracting 60 degrees from 315 degrees, resulting in 255 degrees. It is crucial to distinguish this from merely subtracting a value from a sine calculation, as that can lead to different results.
Here's a quick guide to remember:

• Always resolve operations inside the trigonometric brackets first.
• Once you have the new angle, apply the sine, cosine, etc., to find the trigonometric value at this specific angle.

This distinction is particularly important because confusion can lead to incorrect answers, especially in an exam setting.

Exact Values of Trigonometric Expressions

Finding the exact values of trigonometric expressions often involves utilizing the unit circle or trigonometric tables. Exact values are those expressed in terms of radicals and fractions rather than decimals.
Working with angles like 255° or 315°, you can reference points on the unit circle to determine their sine or cosine values, which are often standard angles extended to compass bearings. For instance:

• \( \sin 255^\circ \) is found to be \(-\sqrt{2}/2\) by understanding its related acute angle and reference within the unit circle.
• \( \sin 315^\circ \) also yields \(-\sqrt{2}/2\).

These exact values are significant as they provide precise and accurate results when input into further calculations, avoiding rounding errors associated with decimal approximations.

Angle Subtraction Formulas

Angle subtraction formulas, specifically in trigonometry, are used to simplify the calculation of trigonometric functions of angles like \( \sin(A - B) \). This can be particularly useful in reducing more complex angles into sums or differences of known angles.
The formula for sine in terms of subtraction is: \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]This formula can be applied directly if you have or can readily find the sine and cosine values for angles A and B. For example:

• If we were to apply this to 315° and 60°, it would involve recognizing their sine and cosine values from the unit circle and substituting them into the formula accordingly.
• This formula reduces the potential for error by relying on well-known values of standard angles rather than performing potentially error-prone calculations from scratch.

Understanding and correctly applying these formulas help streamline solving trigonometric problems efficiently.