Problem 23 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\sin 13 \pi \quad\) (b) \(\cos 14 \pi \quad\) (c) \(\tan 15 \pi\)

Step-by-Step Solution

Answer

(a) 0; (b) 1; (c) 0.

Step 1: Understand Periodicity of Sine and Cosine

The sine and cosine functions are periodic with a period of \(2\pi\). This means that \(\sin(x + 2k\pi) = \sin x\) and \(\cos(x + 2k\pi) = \cos x\) for any integer \(k\). This property helps in reducing any angle to its co-terminal angle within the standard unit circle range \([0, 2\pi)\).

Step 2: Simplify Given Angles

For (a) \(\sin(13\pi)\): Since the period is \(2\pi\), express \(13\pi\) in terms of \(2\pi\): \(13\pi = 6(2\pi) + \pi\), so \(\sin(13\pi) = \sin(\pi)\). For (b) \(\cos(14\pi)\): Similarly, \(14\pi = 7(2\pi)\), so \(\cos(14\pi) = \cos(0)\). For (c) \(\tan(15\pi)\): \(15\pi = 7(2\pi) + \pi\), so \(\tan(15\pi) = \tan(\pi)\).

Step 3: Evaluate Trigonometric Functions

Evaluate the simplified trigonometric functions using standard unit circle values:For (a) \(\sin(\pi) = 0\).For (b) \(\cos(0) = 1\).For (c) since \(\tan(\pi) = 0\) (tangent of \(\pi\) is zero).

Key Concepts

Periodicity

Periodicity is a vital concept in understanding trigonometric functions. It means that a function repeats its values at regular intervals or periods. Both the sine and cosine functions have a period of \(2\pi\). This implies that their values repeat every \(2\pi\) units. For a frequency angle, we adjust it to its equivalent angle within the range from \(0\) to \(2\pi\).

• The repeated cycle of \(2\pi\) allows easy simplification of trigonometric functions for larger angles.
• For sine and cosine, \(f(x + 2k\pi) = f(x)\), where \(k\) is an integer.

Understanding periodicity helps break down complex angle calculations into simpler ones by reducing angles to their co-terminal angles. Using this concept, you can determine that \( \sin(13\pi) \) reduces to \( \sin(\pi) \), and similar reductions apply for the cosine of \(14\pi\) and the tangent of \(15\pi\).

Unit Circle

The unit circle is a perfect tool to understand the sine, cosine, and tangent functions. It is a circle with a radius of one, centered at the origin of a coordinate plane.

• Angles in the unit circle are measured in radians, starting from the positive x-axis and moving counterclockwise.
• Sine of an angle is the y-coordinate of the point where the corresponding radius intersects the unit circle.
• Cosine of an angle is the x-coordinate of the same point.
• The tangent function is given by the ratio \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

For example, on the unit circle:- \( \sin(\pi) \) corresponds to \(0\) because this point lies on the x-axis where the y-coordinate is zero.- \( \cos(0) \) is \(1\) as it lies on the far right of the circle where the x-coordinate is one.- \( \tan(\pi) \) is \(0\), following the ratio of sine over cosine.

Exact Values

Exact values refer to the specific trigonometric values that can be calculated using the unit circle for standard angles. These values are useful because they provide accurate results without approximation.

• The sine, cosine, and tangent of standard angles such as \(0, \pi/2, \pi, 3\pi/2, \) and \(2\pi\) are well known and frequently used in trigonometry problems.
• The exact values for sine and cosine involve fractions \(-1, -\frac{\sqrt{2}}{2}, 0, \frac{\sqrt{2}}{2}, 1\) depending on the angle.
• The tangent function, being the ratio of sine and cosine, takes values like \(0, 1, -1, \) or undefined at these notable positions.

By deriving the exact values from the unit circle, problems such as finding \( \sin(13\pi) \), \( \cos(14\pi) \), and \( \tan(15\pi) \) become straightforward calculations. Using the properties of periodicity and unit circle positions, these values are determined to be \(0\), \(1\), and \(0\), respectively.