Problem 26 - 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. $$ t=-2 \pi / 3 $$

Step-by-Step Solution

Answer

Reference angle is \(\pi/3\), \(\sin t = -\frac{\sqrt{3}}{2}\), \(\cos t = -\frac{1}{2}\).

Step 1: Convert Negative Angle to Positive Equivalent

To find a reference angle, first express the given negative angle, \( t = -\frac{2\pi}{3} \), in its positive equivalent by adding \( 2\pi \). This ensures that the angle is within the range of \( 0 \) to \( 2\pi \). Thus, \( t + 2\pi = -\frac{2\pi}{3} + \frac{6\pi}{3} = \frac{4\pi}{3} \).

Step 2: Determine Reference Angle

Find the reference angle \( t^{\prime} \) by determining the acute angle formed with the x-axis. Since \( \frac{4\pi}{3} \) lies in the third quadrant, the reference angle is found by \( t^{\prime} = \pi - (\frac{4\pi}{3} - \pi) = \frac{\pi}{3} \).

Step 3: Calculate \( \sin t \)

In the third quadrant, sine is negative. The reference angle’s sine is \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\). Thus, \( \sin t = -\sin(t^{\prime}) = -\frac{\sqrt{3}}{2} \).

Step 4: Calculate \( \cos t \)

In the third quadrant, cosine is also negative. The reference angle’s cosine is \(\cos(\frac{\pi}{3}) = \frac{1}{2}\). Hence, \( \cos t = -\cos(t^{\prime}) = -\frac{1}{2} \).

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that relate different trigonometric functions to one another. They allow us to simplify complex expressions and solve equations. Some basic identities include the Pythagorean identity:

• \( \sin^2\theta + \cos^2\theta = 1 \)
• Reciprocal identities like \( \sin\theta = \frac{1}{csc\theta} \)
• Quotient identities such as \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)

These identities are crucial when evaluating trigonometric functions of reference angles, and understanding them helps in recognizing angle patterns.
For example, given the reference angle \( \frac{\pi}{3} \), we use the identity for sine: \( \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \). This allows us to find sine values across different quadrants by adjusting the sign according to the identity and position in the unit circle.

Converting Negative Angles

Negative angles in trigonometry can be tricky, but they simply indicate a direction of rotation. While a positive angle means counterclockwise rotation from the x-axis, a negative angle means clockwise rotation.

• To convert a negative angle to its positive equivalent, we add \( 2\pi \) (one full rotation) to the angle until it becomes positive.
• For \( t = -\frac{2\pi}{3} \), adding \( 2\pi \) gives \( t + 2\pi = \frac{4\pi}{3} \).
• This transforms our negative angle into something more manageable and places it within the standard interval \( [0, 2\pi] \).

This conversion is key for determining the proper reference angle, which will help us analyze its sine and cosine in the circle's quadrants.

Quadrants of the Unit Circle

The unit circle is a powerful concept in trigonometry, helping us understand how different angles relate to trigonometric functions. It's a circle with a radius of 1, centered at the origin of the coordinate plane. The unit circle is divided into four quadrants, each influencing the sign of

• sine
• cosine
• tangent functions

Here's a brief overview:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive; cosine is negative.
• Third Quadrant: Both sine and cosine are negative, as seen with \( \frac{4\pi}{3} \).
• Fourth Quadrant: Sine is negative; cosine is positive.

Understanding which quadrant an angle lies in helps determine the signs of its trigonometric functions.
For \( \frac{4\pi}{3} \), the angle is in the third quadrant, where both \( \sin \) and \( \cos \) are negative. This knowledge, combined with the reference angle \( \frac{\pi}{3} \), allows us to accurately calculate \( \sin t \) and \( \cos t \).