Problem 24 - Precalculus with Calculus Previews

Question

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=-11 \pi / 6 $$

Step-by-Step Solution

Answer

Reference angle \(t' = \frac{\pi}{6}\); \(\sin t = \frac{1}{2}\), \(\cos t = \frac{\sqrt{3}}{2}\).

Step 1: Determine the equivalent angle in the standard position

Convert the negative angle to a positive angle in standard position by adding a full circle \(2\pi\). This is calculated as follows:\[ t = -\frac{11\pi}{6} + 2\pi = -\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{\pi}{6}.\]Thus, the equivalent angle in standard position is \(\frac{\pi}{6}\).

Step 2: Identify the reference angle

The reference angle \(t^{\prime}\) is the smallest positive angle that is equivalent to \(t\) obtained from Step 1. Since \(\frac{\pi}{6}\) is already a positive acute angle (less than \(\pi/2\)), it serves as its own reference angle:\[t^{\prime} = \frac{\pi}{6}.\]

Step 3: Calculate \(\sin t\) and \(\cos t\)

We use the reference angle \(t^{\prime} = \frac{\pi}{6}\) to find the values of \(\sin\) and \(\cos\), since trigonometric functions of an angle and its reference angle are the same in absolute value.From trigonometric identities, for \(\frac{\pi}{6}\):\[\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\]and\[\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}.\]

Key Concepts

Reference Angle

A reference angle is crucial for understanding the properties of angles in trigonometry, especially those not in the first quadrant. It is the smallest positive acute angle that an original angle makes with the x-axis. This concept helps simplify calculations for trigonometric functions.

To find the reference angle, first ensure the angle is in standard position, which means its terminal side is rotated counter-clockwise from the positive x-axis. In the case of a negative angle, you convert it to a positive equivalent by adding full rotations of the circle, which equals to \(2\pi\) radians for each rotation.

For example, for \( t = -\frac{11\pi}{6} \), adding \(2\pi\) results in \( \frac{\pi}{6} \), which is already an acute angle less than \(\pi/2\). Thus, it serves as its own reference angle. Generally, the reference angle is found using the formulas for the different quadrants, but in this case, it is straightforward because the angle is already acute.

Sine and Cosine Values

Trigonometric functions, such as sine and cosine, are often tied to reference angles because their values depend solely on this acute angle in any quadrant.

The values of sine and cosine for some common reference angles are memorized due to their frequent use. For \( \frac{\pi}{6} \), these values are:

• \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)
• \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)

These values remain the same in absolute terms across all quadrants but their signs vary depending on where the angle's terminal side lies.

In quadrants II and III, for instance, sine is positive, while in quadrants III and IV, cosine is negative. This is because of the direction the angle's extension goes with respect to the x and y-axes. For \( t = -\frac{11\pi}{6} \), its reference angle \( \frac{\pi}{6} \) tells us that the sine is positive while cosine remains positive as the angle lies within the first quadrant.

Standard Position Angle

An angle in standard position is one whose initial arm is positioned along the positive x-axis, and whose rotation reaches the final position by moving either clockwise (negative angle) or counter-clockwise (positive angle).

The full revolution of a circle in radians is \(2\pi\), equal to 360 degrees. To convert negative angles into this position, like in the case of \( -\frac{11\pi}{6} \), you add \(2\pi\) to find its equivalent angle. Here, it adds to \( \frac{\pi}{6} \). This angle is more manageable for analyzing the sine and cosine functions.

By converting an angle into standard position, you ensure the use of consistent reference points for polar coordinates. This facilitates understanding how angles interact and ultimately helps simplify and communicate concepts in trigonometry more effectively.