Problem 14 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \left(-\frac{3 \pi}{2}\right)\) (b) \(\cos \left(-\frac{3 \pi}{2}\right)\) (c) \(\cot \left(-\frac{3 \pi}{2}\right)\)

Step-by-Step Solution

Answer

(a) -1, (b) 0, (c) 0

Step 1: Understanding the Angle

The given angle is \(-\frac{3\pi}{2}\). This is equivalent to rotating clockwise through \(-270\) degrees (since \(-\frac{3\pi}{2} \times \frac{180}{\pi} = -270\) degrees). In the unit circle, this lands at the positive \(+Y\)-axis position.

Step 2: Calculating \\(\sin(-\frac{3\pi}{2})\\)

The \(\sin\) function represents the \(Y\)-coordinate of a point on the unit circle. At \(-\frac{3\pi}{2}\), the point on the unit circle is \((0, -1)\). Hence, \(\sin(-\frac{3\pi}{2}) = -1\).

Step 3: Calculating \\(\cos(-\frac{3\pi}{2})\\)

The \(\cos\) function represents the \(X\)-coordinate of a point on the unit circle. At \(-\frac{3\pi}{2}\), the point is \((0, -1)\). Therefore, \(\cos(-\frac{3\pi}{2}) = 0\).

Step 4: Calculating \\(\cot(-\frac{3\pi}{2})\\)

The \(\cot\) function is defined as \(\frac{\cos(\theta)}{\sin(\theta)}\). Using our previous results: \(\cot(-\frac{3\pi}{2}) = \frac{0}{-1} = 0\).

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. The unique property of the unit circle is that every point on the circle corresponds to an angle, and the coordinates of these points are the values of the sine and cosine functions for that angle.

The coordinates \((x, y)\) of any point on the unit circle can be represented as \((cos(\theta), sin(\theta))\). This representation helps us visualize the angle and compute trigonometric values easily. Additionally, because the radius is 1, we can use the Pythagorean identity: \(\cos^2(\theta) + \sin^2(\theta) = 1\).

The unit circle allows us to determine the signs of trigonometric functions by the quadrant they fall in:

• Quadrant I: all trigonometric values are positive.
• Quadrant II: sine is positive, cosine and tangent are negative.
• Quadrant III: tangent is positive, sine and cosine are negative.
• Quadrant IV: cosine is positive, sine and tangent are negative.

Understanding these basic quadrant rules and the properties of the unit circle is key to mastering trigonometry.

Angles in Radians

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians relate the angle to the actual arc length on the unit circle.

A full circle in radians is \(2\pi\), equivalent to 360 degrees. Therefore, \(-\frac{3\pi}{2}\) radians can be converted into degrees by multiplying it by \(\frac{180}{\pi}\). This conversion indicates a \-270-degree rotation around the circle, which helps us understand its positioning in reference to the unit circle.

Using radians has mathematical advantages:

• They simplify the calculation of angles.
• They are inherent in calculus and more natural in mathematical expressions.

Learning to work with radians will hence enhance your understanding of trigonometry and its applications across different mathematical fields.

Exact Values of Trigonometric Functions

Exact trigonometric values refer to precise, unapproximated values of sine, cosine, tangent, and other trigonometric functions at specific angles. Using the unit circle, we can pinpoint these values for angles that correspond to key points, such as \(\pi/6, \pi/4, \pi/3\), and more.

For example, at \(-\frac{3\pi}{2}\), which equals \(-270\) degrees, we locate the point on the unit circle at \(0, -1\). These coordinates provide the direct values for sine and cosine:

• \(\sin(\theta) = -1\)
• \(\cos(\theta) = 0\)

Since cotangent \(\cot(\theta)\) is defined as \(\frac{\cos(\theta)}{\sin(\theta)}\), \(\cot(-\frac{3\pi}{2})\) results in 0.

Exact values are crucial because they give us definitive answers without rounding errors. Recognizing these values also helps you evaluate functions, solve equations, and understand more complex mathematical concepts with confidence.