Problem 26 - Calculus

Question

Calculate the quantity without using the the trigonometric functions on your calculator. You are given that \(\sin (\pi / 12)=0.259\) and \(\cos (\pi / 5)=0.809 .\) You may want to draw a picture showing the angles involved and check your answer on a calculator. $$\cos \frac{\pi}{12}$$

Step-by-Step Solution

Answer

\(\cos\left(\frac{\pi}{12}\right) \approx 0.9659\)

Step 1: Identify the trigonometric identity to use

We know the angles are fractions of common angles in trigonometry. To find \( \cos \left(\frac{\pi}{12}\right) \), let's use the identity \( \cos(2\theta) = 2\cos^2(\theta) - 1 \). This can help us derive the formula we need.

Step 2: Use the identity to express \( \cos(\pi/12) \)

Notice that \( \pi/12 \) can be expressed as half of \( \pi/6 \). Therefore, we can use the half-angle identity: \( \cos(\pi/6) = \sqrt{\frac{1 + \cos(\pi/3)}{2}} \). Since \( \cos(\pi/3) = 0.5 \), \( \cos(\pi/6) \) can then be calculated.

Step 3: Calculate \(\cos(\pi/6)\) using known values

Using the value \( \cos(\pi/3) = 0.5 \), plug it into the half-angle identity to get: \( \cos(\pi/6) = \sqrt{\frac{1 + 0.5}{2}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}\).

Step 4: Apply double-angle identity for precise calculation

Now, apply the double-angle formula, knowing that \(\cos\left(\frac{\pi}{6}\right) = \cos(2 \times \frac{\pi}{12}) \) is related to \(\cos\left(\frac{\pi}{12}\right)\) by the identity \( \cos\left(2\theta\right) = 1 - 2\sin^2\left(\theta\right) \). We've already know \(\sin\left(\frac{\pi}{12}\right) = 0.259\).

Step 5: Use the known sine value to find cosine

Given \( \sin\left(\frac{\pi}{12}\right) = 0.259 \), compute \(\cos\left(\frac{\pi}{12}\right) = \sqrt{1 - \sin^2\left(\frac{\pi}{12}\right)}\). Substitute the value: \( \cos\text = \sqrt{1 - (0.259)^2} \approx \sqrt{1 - 0.067081} \approx \sqrt{0.932919} \).

Step 6: Calculate final result and verify

Calculate \( \sqrt{0.932919} \) to approximate \(\cos\left(\frac{\pi}{12}\right)\). Doing this calculation gives \( \cos\left(\frac{\pi}{12}\right) \approx 0.9659\). Verify via calculation, using a calculator to confirm if needed.

Key Concepts

Cosine Function

In trigonometry, the cosine function is one of the primary trigonometric ratios. It relates the adjacent side of a right triangle to the hypotenuse. It allows us to measure the angular relationships and properties without direct measuring. The cosine function is periodic, with a period of \(2\pi\), which means it repeats every \(2\pi\) radians.

• Cosine values range from -1 to 1, making them very handy in defining the unit circle, where they represent the x-coordinate of a point.
• The cosine of an angle is often used in conjunction with the sine and tangent functions to solve various trigonometric equations or expressions.

When calculating \(\cos\left(\frac{\pi}{12}\right)\), knowing certain cosine values, such as \(\cos\left(\frac{\pi}{3}\right) = 0.5\), helps in applying identities like the half-angle identity to find unknown values.

Half-Angle Identity

The half-angle identity offers a way to find the trigonometric values of half of a given angle. It is especially useful when dealing with angles that are not standard angles on the unit circle.
For cosine, the half-angle formula can be expressed as:
\( \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}} \).
This equation allows us to calculate \(\cos\left(\frac{\pi}{12}\right)\) by expressing it in terms of \(\pi/6\).

• It's essential to choose the correct sign of the square root based on the quadrant where the angle lies.
• Since \(\frac{\pi}{12}\) is in the first quadrant, this ensures the cosine value is positive.

Remember, applying the half-angle identity involves knowledge of other angle values to backfill into the identity smoothly.

Double-Angle Identity

The double-angle identity is fundamental in simplifying expressions in trigonometry. This identity relates the trigonometric functions of double an angle, making it a powerful tool in solving equations.
For cosine, it is expressed as:
\(\cos(2\theta) = 2\cos^2(\theta) - 1\) or \(\cos(2\theta) = 1 - 2\sin^2(\theta)\).
These forms allow flexibility depending on whether sine or cosine values are easier to use or given in a problem.

• This identity can transform a trigonometric equation to achieve a form where known angles or computed values can be used easily.
• In the exercise, \(\cos\left(\frac{\pi}{6}\right)\) was calculated as \(\cos(2 \times \frac{\pi}{12})\), illustrating its application for precise computations.

Finally, by substituting the available \(\sin\left(\frac{\pi}{12}\right)\) value, it's straightforward to derive \(\cos\left(\frac{\pi}{12}\right)\), affirming the practical utility of the double-angle identity in the problem.