Problem 101 - Algebra and Trigonometry

Question

Find the exact value of \(\sin \frac{7 \pi}{12}\).

Step-by-Step Solution

Answer

The exact value of sin(7π/12) is \(\frac{\sqrt{6} - \sqrt{2}}{4} \).

Step 1 - Recognize the Required Trigonometric Function

The problem requires finding the exact value of \(\text{sin} \frac{7 \pi}{12}\).

Step 2 - Use Angle Addition Formula

Rewrite \(\frac{7\pi}{12}\) as the sum of two angles whose sine values we know with exact precision. Observe that \(\frac{7\pi}{12} = \frac{3\pi}{4} + \frac{\pi}{6}\).

Step 3 - Apply Angle Addition Formula for Sine

Use the angle addition formula for sine: \[ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \]. In our case, \(\frac{7\pi}{12}\) can be split into \(\frac{3\pi}{4} = a \) and \( \frac{\pi}{6} = b \).

Step 4 - Substitute Known Values of Trigonometric Functions

Substitute \(\frac{3\pi}{4}\) and \( \frac{\pi}{6}\) into the angle addition formula: \[ \sin \left( \frac{3 \pi}{4} + \frac{\pi}{6} \right) = \sin \left( \frac{3 \pi}{4} \right) \cos \left( \frac{\pi}{6} \right) + \cos \left( \frac{3 \pi}{4} \right) \sin \left( \frac{\pi}{6} \right) \].

Step 5 - Calculate the Individual Trigonometric Values

Find the values of \(\sin \left( \frac{3 \pi}{4} \right)\), \(\cos \left( \frac{3 \pi}{4} \right)\), \(\sin \left( \frac{\pi}{6} \right)\), and \(\cos \left( \frac{\pi}{6} \right)\): \[ \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} \] \[ \cos \left( \frac{3 \pi}{4} \right) = -\frac{\sqrt{2}}{2} \] \[ \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \] \[ \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \].

Step 6 - Substitute the Values into the Formula

Perform the substitutions: \[ \sin \left( \frac{3 \pi}{4} + \frac{\pi}{6} \right) = \left( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \right) + \left( -\frac{\sqrt{2}}{2} \cdot \frac{1}{2} \right) \].

Step 7 - Simplify the Expression

Combine the values to simplify: \[ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \].

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in understanding angles and their relationships in geometry. They stem from the basic right triangle and circle principles. Here are some key trigonometric functions:

• Sine (\text{sin}): This function relates the opposite side to the hypotenuse of a right-angled triangle. For example, in a triangle with sides 'opposite' and 'hypotenuse', \text{sin}(\theta) = \( \frac{opposite}{hypotenuse} \).
• Cosine (\text{cos}): This function relates the adjacent side to the hypotenuse. \text{cos}(\theta) = \( \frac{adjacent}{hypotenuse} \).
• Tangent (\text{tan}): This is the ratio of sine and cosine: \text{tan}(\theta) = \( \frac{sin(\theta)}{cos(\theta)} \).

Understanding these functions is crucial. They help solve many problems involving angles and distances. For instance, finding the exact value of \text{sin} \( \frac{7 \pi}{12} \) requires a good grasp of these concepts. Exploring these properties will save you a lot of time and make solving trigonometric problems easier.

Angle Addition Formula

The angle addition formula is a powerful tool in trigonometry. It allows you to calculate the sine, cosine, or tangent of a sum or difference of two angles. In our case, we use the sine angle addition formula.

The formula is:
\( \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \)

For example, to find \( \sin \left( \frac{7 \pi}{12} \right) \), we can rewrite it as the sum of \( \frac{3 \pi}{4} \) and \( \frac{\pi}{6} \):
\( \sin \left( \frac{7 \pi}{12} \right) = \sin \left( \frac{3 \pi}{4} + \frac{\pi}{6} \right) \).
Using the formula:
\( \sin \left( \frac{3 \pi}{4} + \frac{\pi}{6} \right) = \sin \left( \frac{3 \pi}{4} \right) \cos \left( \frac{\pi}{6} \right) + \cos \left( \frac{3 \pi}{4} \right) \sin \left( \frac{\pi}{6} \right) \)

This formula breaks down the problem into smaller, manageable parts, leveraging known values of trigonometric functions. It simplifies complex expressions and makes seemingly difficult problems more approachable.

Simplifying Expressions

Simplifying expressions is an essential skill, especially in trigonometry. Once we've applied our formulas and performed substitutions, the next step is to simplify the resulting expressions. This process makes it possible to find exact values.

After substituting known trigonometric values into the angle addition formula:
\( \sin \left( \frac{3 \pi}{4} + \frac{\pi}{6} \right) = \left( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \right) + \left( -\frac{\sqrt{2}}{2} \cdot \frac{1}{2} \right) \).

We can simplify each part:
\( \frac{\sqrt{2} \cdot \sqrt{3}}{4} = \frac{\sqrt{6}}{4} \)
\( -\frac{\sqrt{2} \cdot 1}{4} = -\frac{\sqrt{2}}{4} \)

Combine these results:
\( \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \).

Remember: simplifying requires careful combining of like terms and understanding how to manipulate radical expressions. This ensures your final answer is in its simplest and most exact form. Making accurate simplifications will lead to correct answers and a deeper comprehension of trigonometric relationships.