Problem 11 - Precalculus with Calculus Previews

Question

Find the exact value of (a) \(\sin t\) and (b) \(\cos t\) for the given value of \(t\). Do not use a calculator. $$ t=-\pi / 2 $$

Step-by-Step Solution

Answer

\( \sin t = -1 \) and \( \cos t = 0 \) for \( t = -\pi / 2 \).

Step 1: Identify the Angle

For the given problem, the angle is defined as \( t = -\pi / 2 \). This angle is measured in radians and corresponds to the negative direction along the unit circle.

Step 2: Locate the Angle on the Unit Circle

The angle \( -\pi / 2 \) is a standard position angle on the unit circle. Starting from the positive x-axis, move clockwise (-) by \( \pi / 2 \) radians. This angle points to the negative y-axis.

Step 3: Evaluate \( \sin t \)

On the unit circle, the \( y \)-coordinate corresponds to \( \sin t \). For \( t = -\pi / 2 \), the point on the unit circle is \((0, -1)\). Thus, \( \sin \left(-\frac{\pi}{2}\right) = -1 \).

Step 4: Evaluate \( \cos t \)

On the unit circle, the \( x \)-coordinate corresponds to \( \cos t \). For \( t = -\pi / 2 \), the point on the unit circle is \((0, -1)\). Thus, \( \cos \left(-\frac{\pi}{2}\right) = 0 \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of a triangle to the lengths of its sides. They are fundamental in the realm of geometry and have applications across various fields such as physics, engineering, and music theory. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each function has a crucial role on the unit circle. The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. This circle can be used to define trigonometric functions for all angles, not just those between 0 and 90 degrees.

• Sine (\( \sin \)): Represents the y-coordinate of a point on the unit circle corresponding to any given angle.
• Cosine (\( \cos \)): Corresponds to the x-coordinate of a point on the unit circle representing that angle.
• Tangent (\( \tan \)): A ratio of sine to cosine, thus \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Radians

Radians are a way of measuring angles, different from degrees. One radian is the angle formed when the arc length of a circle equals the radius of the circle. This measurement correlates directly with the unit circle. The whole circle measures \(2\pi\) radians, making it a complete rotation. Common angle conversions in radians include:

• 90 degrees: \(\frac{\pi}{2}\) radians
• 180 degrees: \(\pi\) radians
• 360 degrees: \(2\pi\) radians

Using radians simplifies many mathematical calculations and expressions, making them a preferred choice in advanced mathematics. Additionally, the seamless integration of radians into calculus because of their natural derivation from circle geometry is another reason why they are broadly used.

sin t

The function \( \sin t \) calculates the y-coordinate on the unit circle for an angle \( t \). When considering the angle \( t = -\frac{\pi}{2} \), this places you on the circle's negative y-axis since a negative sign indicates a clockwise movement. The specific point associated with \( -\frac{\pi}{2} \) radians on the unit circle is \((0, -1)\).
Since sine corresponds to the y-coordinate, we find that \( \sin(-\frac{\pi}{2})= -1 \). This makes \( \sin t \) negative as expected when moving into the lower half of the circle. Knowing how to find \( \sin t \) helps solve trigonometry problems without reaching for a calculator, thus strengthening your understanding of functions and angles.

cos t

The function \( \cos t \) represents the x-coordinate on the unit circle for an angle \( t \). In the problem where \( t = -\frac{\pi}{2} \), you end up directly at the bottom of the circle on the unit circle. Since this position is represented by the point \((0, -1)\), the x-coordinate is 0.
Therefore, \( \cos(-\frac{\pi}{2}) = 0 \). Cosine's behavior is unique depending on which quadrant of the unit circle you are in. In this particular scenario, the negative-x direction does not play a role, making the x-component zero. Understanding cosine this way lets you determine angle coordinates quickly and aids in predicting waveforms and oscillations in physics and engineering contexts.