Problem 18 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Formula for Negative Angles

For any trigonometric function, the formula for negative angles helps simplify the calculation. We have:- \( \sin(-x) = -\sin(x) \)- \( \cos(-x) = \cos(x) \)- \( \tan(-x) = -\tan(x) \) With this in mind, we can apply these formulas to find the exact values of the given expressions.

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

Apply the formula for sine of a negative angle: \( \sin(-x) = -\sin(x) \).So, \( \sin\left(-\frac{3\pi}{2}\right) = -\sin\left(\frac{3\pi}{2}\right) \).The angle \( \frac{3\pi}{2} \) corresponds to 270 degrees, and at this angle, \( \sin(\frac{3\pi}{2}) = -1 \). Thus, \( \sin\left(-\frac{3\pi}{2}\right) = 1 \).

Step 3: Simplify \( \cos(-225^{\circ}) \)

Apply the formula for cosine of a negative angle: \( \cos(-x) = \cos(x) \).So, \( \cos(-225^{\circ}) = \cos(225^{\circ}) \).The angle 225 degrees is in the third quadrant where cosine is negative, and has a reference angle of 45 degrees.Thus, \( \cos(225^{\circ}) = -\cos(45^{\circ}) = -\frac{\sqrt{2}}{2} \).

Step 4: Simplify \( \tan(-\pi) \)

Apply the formula for tangent of a negative angle: \( \tan(-x) = -\tan(x) \).So, \( \tan(-\pi) = -\tan(\pi) \).The angle \( \pi \) corresponds to 180 degrees where the tangent function is zero.Thus, \( \tan(-\pi) = 0 \).

Key Concepts

Negative Angle Identities

Trigonometric identities for negative angles are very useful for simplifying calculations in trigonometry. These identities help us deal with negative inputs for sine, cosine, and tangent functions.

• The sine of a negative angle, \( \sin(-x) \), is equal to the negative of the sine of the angle: \( -\sin(x) \).
• The cosine of a negative angle, \( \cos(-x) \), remains the same as the cosine of the angle: \( \cos(x) \).
• The tangent of a negative angle, \( \tan(-x) \), is equal to the negative of the tangent of the angle: \( -\tan(x) \).

These relative symmetry properties reflect the graphical nature of each function on the coordinate plane. Each function has its distinctive characteristic relative to the x-axis or y-axis. When working with negative angles, these identities allow us to easily convert them into positive ones, which are often easier to evaluate.

Exact Values of Trigonometric Functions

Determining the exact values of trigonometric functions is a crucial skill in mathematics. It involves knowing special angles at specific intervals that give precise values.To find the exact value of a trigonometric function:

• Identify common reference angles, such as 0, 30, 45, 60, and 90 degrees, or their radian equivalents.
• Recognize values like \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \), \( \cos(60^{\circ}) = \frac{1}{2} \), \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \).
• Apply known values using specific identities to compute the desired values accurately.

This process helps in scenarios such as transforming angles within the unit circle or simplifying expressions based on calculated trigonometric results.

Unit Circle

The unit circle is a fundamental concept in trigonometry and defines trigonometric functions geometrically. Plotted in the coordinate plane, the unit circle has a radius of one unit, centered at the origin (0,0).

• Every angle on the unit circle corresponds to a coordinate point \((x, y)\), where \(x\) is the cosine of the angle and \(y\) is the sine of the angle.
• For example, at \(\frac{\pi}{4}\) or 45 degrees, the point is \(\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)\).
• This tool is imperative for visualizing the interplay between angles and the values of sine, cosine, and tangent on a coordinate plane.

By understanding the positions of angles on the unit circle, we can better comprehend why certain identities and expressions hold true and how to apply them in solving problems.

Trigonometric Functions Simplification

Simplifying trigonometric functions is crucial for solving more complex expressions in mathematics and physics.

• It involves the application of trigonometric identities, such as Pythagorean identities or angle sum and difference identities.
• To simplify an expression, always look to substitute equivalent identities that match components of the expression you have.
• Identities like \( \cos^2(x) + \sin^2(x) = 1 \) can also reduce many problems into simpler forms.

Simplification not only makes calculations easier but also provides insight into the behavior of trigonometric functions and their relationships with one another.