Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 17-20: Use a formula for negatives to find the exact value. (a) \(\sin \left(-\frac{3 \pi}{2}\right)\) (b) \(\cos \left(-225^{\circ}\right)\) (c) \(\tan (-\pi)\)

Step-by-Step Solution

Answer

(a) 1, (b) \(-\frac{\sqrt{2}}{2}\), (c) 0.

Step 1: Understanding the Sine Function for Negative Angles

The sine function is an odd function, meaning that \( \sin(-x) = -\sin(x) \). We can use this property to find the sine of a negative angle. Let's apply this to \( \sin \left(-\frac{3\pi}{2}\right) \).

Step 2: Simplifying \( \sin\left(-\frac{3\pi}{2}\right) \)

Using the odd property of sine, we have \( \sin \left(-\frac{3\pi}{2}\right) = -\sin \left(\frac{3\pi}{2}\right) \). We know that \( \sin \left(\frac{3\pi}{2}\right) = -1 \). Thus, \( \sin \left(-\frac{3\pi}{2}\right) = --1 = 1 \).

Step 3: Understanding the Cosine Function for Negative Angles

The cosine function is even, meaning that \( \cos(-x) = \cos(x) \). This property can help us find the cosine of a negative angle such as \( \cos(-225^{\circ}) \).

Step 4: Simplifying \( \cos(-225^{\circ}) \)

Using the even property of cosine, \( \cos(-225^{\circ}) = \cos(225^{\circ}) \). Since \( 225^{\circ} \) is in the third quadrant where cosine is negative, and the reference angle is \( 45^{\circ} \), we have \( \cos(225^{\circ}) = -\cos(45^{\circ}) = -\frac{\sqrt{2}}{2} \). Therefore, \( \cos(-225^{\circ}) = -\frac{\sqrt{2}}{2} \).

Step 5: Understanding the Tangent Function for Negative Angles

The tangent function is also an odd function, meaning \( \tan(-x) = -\tan(x) \). We will apply this to \( \tan(-\pi) \).

Step 6: Simplifying \( \tan(-\pi) \)

Using the odd property of tangent, \( \tan(-\pi) = -\tan(\pi) \). Since \( \tan(\pi) = 0 \), we have \( \tan(-\pi) = -0 = 0 \).

Key Concepts

Sine Function Properties

The sine function, often denoted as \( \sin(\theta) \), has distinct characteristics that are important to understand. One key property of the sine function is that it is an odd function. This means that for any angle \( x \), the sine of the negative angle is the negative of the sine of the angle, expressed as \( \sin(-x) = -\sin(x) \).
This property can be incredibly useful when working with negative angles. For instance, let's look at \( \sin\left(-\frac{3\pi}{2}\right) \). Applying the property of odd functions, we get

• \( \sin\left(-\frac{3\pi}{2}\right) = -\sin\left(\frac{3\pi}{2}\right) \)
• We know \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
• So, \( \sin\left(-\frac{3\pi}{2}\right) = --1 = 1 \)

By recognizing this property, the sine value of any negative angle can be easily determined without the need for further complex calculations.

Cosine Function Properties

The cosine function, denoted as \( \cos(\theta) \), also possesses its own unique properties. Unlike sine, the cosine function is an even function. This means that for any angle \( x \), the cosine of the negative angle is the same as the cosine of the angle, given by \( \cos(-x) = \cos(x) \).
This property is particularly valuable when calculating the cosine of negative angles. Take \( \cos(-225^{\circ}) \) as an example. Using the even characteristics of cosine:

• \( \cos(-225^{\circ}) = \cos(225^{\circ}) \)
• Since \( 225^{\circ} \) is in the third quadrant where cosine is negative, and it has a reference angle of \( 45^{\circ} \), we use \( \cos(225^{\circ}) = -\cos(45^{\circ}) = -\frac{\sqrt{2}}{2} \)

Understanding the even nature of cosine allows for quick and efficient computations of cosine values, especially with negative angles.

Tangent Function Properties

The tangent function, represented as \( \tan(\theta) \), is another trigonometric function that must be considered when dealing with angles. Much like the sine function, the tangent function is an odd function. This means that for any angle \( x \), the tangent of the negative angle is the negative of the tangent of the angle, formulated as \( \tan(-x) = -\tan(x) \).
Considering this property, evaluating \( \tan(-\pi) \) becomes straightforward:

• \( \tan(-\pi) = -\tan(\pi) \)
• Since \( \tan(\pi) = 0 \), we thus have \( \tan(-\pi) = -0 = 0 \)

The understanding that tangent is an odd function simplifies the process of finding tangent values at negative angles, making calculations effortless.

Negative Angles in Trigonometry

Understanding how to work with negative angles is crucial in trigonometry. Negative angles occur as a result of measuring angles in the clockwise direction, or simply subtracting from a circle's complete rotation. Trigonometric functions handle these angles through properties of symmetry and periodicity.
Key points on dealing with negative angles include:

• The sine and tangent functions are odd functions, acknowledging that their values invert with negative angles, i.e., \( \sin(-x) = -\sin(x) \) and \( \tan(-x) = -\tan(x) \).
• The cosine function is even, so the cosine of negative angles remains unchanged, \( \cos(-x) = \cos(x) \).
• Using these properties simplifies the computation of trigonometric values without needing detailed angle conversions or diagrams.

Gaining mastery over these properties will enable smoother problem-solving and a deeper understanding of trigonometric functions in different contexts.