Problem 16 - Algebra 2

Question

Find the exact value of each expression. \(\sin 255^{\circ}\)

Step-by-Step Solution

Answer

\( \sin 255^{\circ} = -\frac{\sqrt{3}}{2} \).

Step 1: Express in Terms of Known Angles

Find an angle that, when added or subtracted from a known angle, results in 255°. Notice that 255° can be expressed as 180° + 75°.

Step 2: Use Angle Sum Formulas

According to the angle sum identity for sine, \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Substitute \( a = 180^{\circ} \) and \( b = 75^{\circ} \).

Step 3: Calculate Known Values

For \( \sin 180^{\circ} = 0 \), \( \cos 180^{\circ} = -1 \).For \( \sin 75^{\circ} = \cos 15^{\circ} \) and \( \cos 75^{\circ} = \sin 15^{\circ} \), use the identities \( \sin 75^{\circ} = \frac{\sqrt{3}}{2} \text{ and } \cos 75^{\circ} = \frac{1}{2} \).

Step 4: Plug in Known Values

Substitute into the formula: \( \sin(180^{\circ} + 75^{\circ}) = \sin 180^{\circ} \cos 75^{\circ} + \cos 180^{\circ} \sin 75^{\circ} \). This becomes \(0 \cdot \frac{1}{2} + (-1) \cdot \frac{\sqrt{3}}{2} \).

Step 5: Simplify

Simplify the expression: The value becomes \(-\frac{\sqrt{3}}{2}\).

Key Concepts

Angle Sum Formula

The angle sum formula is a fundamental tool in trigonometry that helps break down complex trigonometric expressions into simpler components involving sums or differences of angles. The primary formulas are used for sine, cosine, and tangent, which allow calculations such as the sum of two angles:

• The formula for sine is: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
• Similarly, for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).

Understanding these formulas is crucial because they let you express any trigonometric function of a non-standard angle as a combination of angles with known trigonometric values. For instance, by recognizing that \(255^{\circ}\) can be expressed as \(180^{\circ} + 75^{\circ}\), this method allows trigonometric calculations based on known values. This process simplifies calculation and reveals deeper symmetries in trigonometric functions.

Sine Function

The sine function is a periodic function that arises in many different areas of mathematics and physics. It is one of the fundamental trigonometric functions and appears regularly in various calculations:

• The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse.
• In the unit circle, the sine of an angle is the vertical coordinate of the corresponding point on the circle.

It is crucial to remember the key values of sine for common angles, especially those for \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ},\) and \(90^{\circ}\). The behavior of the sine function over different quadrants is also important. For example, the sine of an angle obtained by summing \(180^{\circ}\) with another angle, remains solely influenced by that other angle's properties, adjusted by a factor due to cosine of \(180^{\circ}\).

Exact Trigonometric Values

Exact trigonometric values refer to the precise, often simple values that certain trigonometric functions take at specific standard angles. Common angles (in degrees) are \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ},\) and \(90^{\circ}\). Here are some key facts:

• These angles are often represented using simple fractions or radicals. For example, \(\sin 30^{\circ} = \frac{1}{2}\) or \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\).
• Determining exact values often involves using these standard angles because they simplify the calculations and provide a precise result.

Such values are pivotal for swiftly calculating unknown trigonometric values without needing a calculator, especially in problems like finding the sine of combined angles. Knowing these values helps in quickly evaluating expressions like \(\sin 75^{\circ}\), which can often be broken down into sums or differences of standard angles.

Trigonometric Calculations

Trigonometric calculations involve using trigonometric identities and relationships to solve expressions or equations related to angles. These calculations can become quite complex, but understanding the core principles makes them manageable:

• The most common starting point is simplifying expressions using known values and identities, such as the angle sum formula.
• By simplifying, you can express an angle in terms of addition or subtraction to reveal known reference angles, helping you deduce exact values. For instance, as in the example, rewriting \(\sin 255^{\circ}\) as \(\sin(180^{\circ} + 75^{\circ})\) helps apply the corresponding identity directly.

Let's consider the process: using the identity \(\sin(a + b) = \sin a \cos b + \cos a \sin b\), you can break down the expression into calculable parts using standard values for these reference angles, ultimately leading to a precise result. Practicing these techniques is crucial for mastering trigonometric calculations and directly applying them to problems involving angles in real-world scenarios.