Problem 43 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\sin \frac{\pi}{12}$$

Step-by-Step Solution

Answer

The value of \(\sin \frac{\pi}{12}\) to four decimal places is approximately 0.2588.

Step 1: Understand the Trigonometric Functiom

The sine function, often abbreviated 'sin' is one of the basic functions in trigonometry. We are asked to find the value of \(\sin \frac{\pi}{12}\), where \(\pi\) is a mathematical constant whose approximately value is 3.1416.

Step 2: Use a scientific calculator

While one could evaluate this manually using certain trigonometric identities, for this exercise you are required to use a scientific calculator. Ensure your calculator is in Radian mode, not Degree mode, because we're dealing with radians here.

Step 3: Evaluate the Function

After confirming that the calculator is in the right mode, input \(\sin \frac{\pi}{12}\). Read out the value that's displayed on the screen. Remember, it should be to four decimal places.

Key Concepts

Scientific Calculator Usage

Scientific calculators are essential tools for students, especially when tackling complex mathematical problems. To use a scientific calculator for evaluating trigonometric functions, you must familiarize yourself with its mode settings. With your device in hand, first ensure that it is set to the 'radian mode' when dealing with radian measurements. Locate the mode button, which normally cycles through different settings such as radians (RAD), degrees (DEG), or gradients (GRAD). For exercises like evaluating \( \sin \frac{\pi}{12} \), which are in radians, you must select RAD mode.

Once you've set the correct mode, identify the trigonometric function keys, usually marked 'SIN', 'COS', 'TAN', and their inverses. Enter the value after pressing the corresponding function key. If you're working with fractions like \( \frac{\pi}{12} \), use the division and Pi (\( \pi \)) keys. To achieve precise results, remember to round according to the problem's instructions—in this case, to four decimal places. Ensuring you input values and operators correctly is crucial in avoiding errors.

Sine Function

The sine function, denoted as \( \sin \), is a fundamental trigonometric function that describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. However, its utility extends beyond triangles to model periodic phenomena such as sound and light waves.

In the unit circle context, the sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. The sine function is periodic with a period of \( 2\pi \) radians or 360 degrees, meaning it repeats values in regular intervals. Moreover, it is an odd function, so \( \sin(-\theta) = -\sin(\theta) \), and it has a range of [-1, 1], limiting its output values within this interval.

Radian Mode Evaluation

Evaluating trigonometric functions often requires switching your calculator to radian mode, as some angles are naturally expressed in radians. A radian is an alternative way of measuring angles based on the radius of a circle. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. The full circle has \( 2\pi \) radians, which is equivalent to 360 degrees.

To evaluate \( \sin \frac{\pi}{12} \) in radian mode, first ensure your calculator is set to RAD. Then, input the radian value directly. Understanding the radian measurement is crucial, as it is often used in calculus and higher mathematics, especially where trigonometric functions and their properties are deeply explored.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. They are used to simplify expressions, solve trigonometric equations, and to compute specific values. A fundamental identity is the Pythagorean identity: \( \sin^2\theta + \cos^2\theta = 1 \), which originates from the Pythagorean theorem for right triangles but applies to any angle \( \theta \).

These identities are immensely helpful when you need to transform one trigonometric function into another or to simplify complex trigonometric expressions. Memorizing the basic identities, such as reciprocal identities (e.g., \( \csc\theta = \frac{1}{\sin\theta} \)) and angle sum or difference formulas, can be highly beneficial when you're without a calculator or when the exercise requires an exact value, such as \( \sin(45^\circ) \) or \( \sin(\frac{\pi}{4}) \).