Problem 13 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Find the exact value of each expression. $$ \sin \left(45^{\circ}-30^{\circ}\right) $$

Step-by-Step Solution

Answer

The exact value of \( \sin (45° - 30°) \) is \( \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \)

Step 1: Identify the type of angles given

The angles given are 45° and 30°. These are special angles in the unit circle with known sine and cosine values. The \(\sin 45° = \cos 45° = \frac{\sqrt{2}}{2}\) and \(\sin 30° = \frac{1}{2}\), \(\cos 30° = \frac{\sqrt{3}}{2}\).

Step 2: Substitute the given angles into the difference of angles identity

Plugging the provided angles into the formula \(\sin(a-b) = \sin a \cos b - \cos a \sin b\), we get \(\sin(45° - 30°) = \sin 45° \cos 30° - \cos 45° \sin 30°\).

Step 3: Substitute the sine and cosine of the angles

Next, substitute the sine and cosine of 45° and 30° into the equation. This leads to \(\sin(45° - 30°) = \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} * \frac{1}{2}\)

Step 4: Simplify the expression

Upon simplifying the equation, we reach the final result which is \(\sin(45° - 30°) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\).

Key Concepts

Exact Trigonometric Values

Understanding exact trigonometric values is essential when dealing with trigonometric problems that involve special angles. These values correspond to points on the unit circle and can be found without the need for approximation. For instance, when asked to find the exact value of an expression like \(\sin(45^\circ - 30^\circ)\), it's important to recall the specific sine and cosine values for 45° and 30°.

The angles 45°, 30°, and their counterparts in other quadrants are part of a group of angles with trigonometric values that can be expressed as ratios of integers and square roots. For a 45° angle, both the sine and cosine values are \(\frac{\sqrt{2}}{2}\), while for 30°, the sine value is \(\frac{1}{2}\) and the cosine value is \(\frac{\sqrt{3}}{2}\). Knowing these values by heart allows you to quickly and accurately substitute them into identities and find exact values for more complex expressions.

Unit Circle Special Angles

The unit circle is a powerful tool for understanding trigonometry, and it becomes even more useful when dealing with special angles. These angles, which include 30°, 45°, and 60°, are strategically placed around the unit circle in order to simplify calculations.

On the unit circle, every point along the circumference represents a coordinate pair \((x, y)\) that corresponds to \((\cos\theta, \sin\theta)\) for any angle \(\theta\).

Why Special Angles Matter

The beauty of special angles is that their sine and cosine values are either fractions involving square roots, or common rational numbers, making them easy to remember and work with. For example, at 30° or \(\pi/6\) radians, the coordinates are \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\), representing \(\cos 30^\circ\) and \(\sin 30^\circ\) respectively.

These specific points make calculating trigonometric functions for these angles straightforward, bypassing the need for calculators and allowing for exact answers in symbolic form.

Sine and Cosine Values

The backbone of trigonometry is the understanding of sine and cosine functions. Sine and cosine values for certain angles can be executed from memory because of their regularity and significance in the unit circle.

For any angle \(\theta\), the sine value represents the y-coordinate, and the cosine value represents the x-coordinate of the point on the unit circle associated with \(\theta\). When studying the circle, you'll notice a pattern that the sine of an angle and the cosine of its complementary angle (adding up to 90°) are equal. This is why \(\sin 45^\circ\) and \(\cos 45^\circ\) are the same, both equating to \(\frac{\sqrt{2}}{2}\).

Application in Problems

When given a trigonometric expression involving subtraction or addition of angles, like \(\sin(45^\circ - 30^\circ)\), we apply the sum or difference identities. These formulas are \(\sin(a - b) = \sin a\cos b - \cos a\sin b\) and \(\cos(a + b) = \cos a\cos b - \sin a\sin b\), which combine known sine-cosine values to reach an exact result. Using knowledge of the exact sine and cosine values for special angles, we can solve these problems symbolically, providing clear, precise answers.