Problem 5 - Precalculus with Calculus Previews

Question

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin \frac{17 \pi}{12} $$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\sqrt{6} + \sqrt{2}}{4}\).

Step 1: Identify Suitable Angles

To find the exact value of the function \( \sin \frac{17\pi}{12} \), look for angles whose sum or difference results in \( \frac{17\pi}{12} \) and are known from trigonometric tables. Notice that \( \frac{17\pi}{12} = \frac{4\pi}{3} + \frac{\pi}{4} \).

Step 2: Recall the Sum Formula for Sine

The sine of the sum of two angles \( A \) and \( B \) can be calculated using the formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). This formula will be used to evaluate \( \sin \left( \frac{4\pi}{3} + \frac{\pi}{4} \right) \).

Step 3: Substitute Known Sine and Cosine Values

Substitute \( A = \frac{4\pi}{3} \) and \( B = \frac{\pi}{4} \), for which you know that: \( \sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2} \), \( \cos \frac{4\pi}{3} = -\frac{1}{2} \), \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), and \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 4: Apply Values into Formula

Using the formula from Step 2, calculate \( \sin \left( \frac{4\pi}{3} + \frac{\pi}{4} \right) \):\[ \sin \frac{17\pi}{12} = \sin \frac{4\pi}{3} \cos \frac{\pi}{4} + \cos \frac{4\pi}{3} \sin \frac{\pi}{4} \]Substituting the known values:\[ = \left( -\frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{2}}{2} \right) + \left( -\frac{1}{2} \right) \left( \frac{\sqrt{2}}{2} \right) \]

Step 5: Simplify the Expression

Simplify the expression from Step 4:\[ = -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \]Combine the terms:\[ = -\frac{\sqrt{6} + \sqrt{2}}{4} \]

Step 6: Final Answer

Thus, the exact value of \( \sin \frac{17\pi}{12} \) is \( -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental tools in trigonometry, which deal with the relationships between the angles and sides of triangles. They are crucial for understanding various phenomena in mathematics and physics. The primary trigonometric functions include:

• Sine ( \(\sin\)): Measures the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine ( \(\cos\)): Measures the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
• Tangent ( \(\tan\)): The ratio of sine to cosine, or equivalently, the ratio of the opposite side to the adjacent side in a right triangle.

The sine and cosine functions are periodic, which means they repeat their values in regular intervals, specifically every \(2\pi\) radians.
The use of trigonometric functions extends beyond triangles, such as in waveforms and circular motion. They are integral for solving equations that involve angles and are widely used in engineering and physics.

Exact Values of Trigonometric Expressions

Finding the exact values of trigonometric expressions means calculating trigonometric functions without approximations or using a calculator. This process often involves using specific angles whose trigonometric values are well-known and have been tabulated.
Some common angles and their trigonometric values include:

• \(30^\circ\) or \(\frac{\pi}{6}\): \(\sin = \frac{1}{2}\), \(\cos = \frac{\sqrt{3}}{2}\)
• \(45^\circ\) or \(\frac{\pi}{4}\): \(\sin = \cos = \frac{\sqrt{2}}{2}\)
• \(60^\circ\) or \(\frac{\pi}{3}\): \(\sin = \frac{\sqrt{3}}{2}\), \(\cos = \frac{1}{2}\)

Memorizing these specific values enables us to solve trigonometric expressions involving these angles exactly. While modern calculators provide numerical approximations, knowing exact values is essential in fields like engineering and physics, where precision is critical.

Angle Addition Identities

Angle addition identities are formulas that help us find the trigonometric values of sums or differences of angles. These identities are pivotal for exploring complex trigonometric problems.
The angle addition identities for sine and cosine are:

• Sine Addition Identity: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
• Cosine Addition Identity: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)

These formulas allow us to break down complex angles into simpler components whose trigonometric values are known. For example, the expression \(\sin(\frac{17\pi}{12})\) can be evaluated using the sine addition identity with known values of sine and cosine at angles \(\frac{4\pi}{3}\) and \(\frac{\pi}{4}\).
Using these identities effectively demands practice and familiarity with trigonometric expressions and their exact values. These identities prove crucial in various mathematics fields, particularly in calculus and analytic geometry.