Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{\pi}{12}$$

Step-by-Step Solution

Answer

The exact values are \(\sin(\frac{\pi}{12}) = \frac{\sqrt{6}-\sqrt{2}}{4}\), \(\cos(\frac{\pi}{12}) = \frac{\sqrt{6}+\sqrt{2}}{4}\), and \(\tan(\frac{\pi}{12})= 2-\sqrt{3}\).

Step 1: Identifying Similar Angles

The first step is to realize that the given angle \(\frac{\pi}{12}\) can be achieved by subtracting two common angles: \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\). Express \(\frac{\pi}{12}\) as such, \(\frac{\pi}{12} = \frac{\pi}{4} - \frac{\pi}{6}\). Using the formula for subtracting cosines, \(\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)\), we get for cosine the following.

Step 2: Apply Cosine Formula

Next, we apply the formula for the cosine of an angle, which is given by \(\cos(\frac{\pi}{12}) = \cos(\frac{\pi}{4} - \frac{\pi}{6}) = \cos(\frac{\pi}{4})\cos(\frac{\pi}{6}) + \sin(\frac{\pi}{4})\sin(\frac{\pi}{6})\). By evaluating it we get, \(\cos(\frac{\pi}{12}) = \frac{\sqrt{2}+1}{2\sqrt{2}} = \frac{\sqrt{6}+\sqrt{2}}{4}\).

Step 3: Apply Sine Formula

Next, apply the sine formula. This is similarly given by \(\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)\). So, \(\sin(\frac{\pi}{12}) = \sin(\frac{\pi}{4} - \frac{\pi}{6}) = \sin(\frac{\pi}{4})\cos(\frac{\pi}{6}) - \cos(\frac{\pi}{4})\sin(\frac{\pi}{6})\). By evaluating it, we get \(\sin(\frac{\pi}{12}) = \frac{\sqrt{2}-1}{2\sqrt{2}} = \frac{\sqrt{6}-\sqrt{2}}{4}\).

Step 4: Calculate Tangent Value

The final value we need is the tangent of the angle. This is given by \(\tan(a) = \frac{\sin(a)}{\cos(a)}\). Plugging in our results from the earlier steps, we get \(\tan(\frac{\pi}{12}) = \frac{\sin(\frac{\pi}{12})}{\cos(\frac{\pi}{12})} = \frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}} = 2-\sqrt{3}\).

Key Concepts

Half-angle formulas

Half-angle formulas are useful tools in trigonometry that help find the values of sine, cosine, and tangent for angles that are fractions, like \( \frac{\pi}{12} \). These formulas come handy when the angle we are dealing with is not standard, and we can relate it to known angles.

The half-angle formulas allow us to express these trigonometric functions using expressions derived from double angles. The formulas for sine, cosine, and tangent of a half-angle \( \frac{\theta}{2} \) are:

• \( \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \)
• \( \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \)
• \( \tan \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \)

The signs depend on the quadrant in which \( \frac{\theta}{2} \) lies.

Exact values of trigonometric functions

Finding exact values of trigonometric functions is an essential step in solving many problems. These values are typically associated with angles that are commonly found in the unit circle, such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \) and so on.

For instance, knowing the exact values of \( \sin \left( \frac{\pi}{4} \right) \) and \( \cos \left( \frac{\pi}{4} \right) \), both \( \frac{\sqrt{2}}{2} \), enables you to easily calculate other related angles through trigonometric identities.

These values are fundamental because they allow us to avoid approximations when calculating further expressions, providing precision in solutions and helping deepen understanding of trigonometric relations.

Angle subtraction identities

Angle subtraction identities are powerful tools for breaking down complex angles into simpler components. Specifically, these identities help express functions like sine, cosine, and tangent for the difference of two known angles.

The formulas are as follows:

• \( \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \)
• \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \)
• \( \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} \)

By using these identities, you can transform a challenging problem into more manageable parts, making complicated calculations much simpler. It's like having a shortcut to find trigonometric values for uncommon angles by using known angles.

Trigonometric problem solving

Trigonometric problem solving involves using various trigonometric identities and formulas to find unknown values or simplify expressions. By understanding and applying these identities, you can solve complex problems step-by-step.

The process usually involves recognizing patterns or familiar angles, applying identities like angle subtraction identities or half-angle formulas, and using known values to find solutions.

For example, if given \( \frac{\pi}{12} \), one might first express this angle in terms of known angles, such as \( \frac{\pi}{4} \) and \( \frac{\pi}{6} \), before applying the necessary identities. Such an approach underlines many trigonometry problems, turning potentially perplexing problems into more digestible calculations.