Problem 38 - Precalculus : Building Concepts and Connections

Question

Find the exact values of all the trigonometric functions for the giocn calues of \(t .\) If a certain value is undefined, state sa Do not use a calculator. $$t=-\frac{14 \pi}{3}$$

Step-by-Step Solution

Answer

Therefore, for \(t= -14\pi/3\), \(\sin(t) = \sqrt{3}/2\), \(\cos(t) = 1/2\), \(\tan(t) = \sqrt{3}\), \(\csc(t) = 2/\sqrt{3}\), \(\sec(t) = 2\), and \(\cot(t) = 1/\sqrt{3}\).

Step 1: Simplify The Angle

Since -14π/3 is not in the range of the unit circle (0 to 2π), convert it to an equivalent angle within the circle by adding multiples of 2π (or 2πn where n is an integer) until the angle falls between 0 and 2π. The result is \(t = -14\pi/3 + 2(5\pi) = π/3\).

Step 2: Evaluate The Basic Trigonometric Functions

Using the unit circle, find that \(\cos(\pi/3) = 1/2\), and \(\sin(\pi/3) = \sqrt{3}/2\). The tangent function is found by \(\tan(\pi/3) = \sin(\pi/3)/\cos(\pi/3) = \sqrt{3}\).

Step 3: Evaluate The Reciprocal Trigonometric Functions

If a function’s value isn’t 0, you can find the reciprocal function’s value by taking the reciprocal of the original function’s value. This leads to \(\sec(\pi/3) = 1/\cos(\pi/3) = 2\), \(\csc(\pi/3) = 1/\sin(\pi/3) = 2/\sqrt{3}\), and \(\cot(\pi/3) = 1/\tan(\pi/3) = 1/\sqrt{3}\).

Key Concepts

Unit Circle

The Unit Circle is a fundamental tool in trigonometry used to define the sine and cosine functions. It is a circle with a radius of one, centered at the origin of a coordinate plane. In this circle:

• The angle is measured in radians, typically ranging from 0 to 2π.
• Every position on the circle corresponds to an angle that emerges from the positive x-axis.
• The coordinates of a point on the circle give the values of cosine and sine for that angle, i.e., if the angle is θ, the coordinates (\( ext{cos}(θ), ext{sin}(θ)\)).

By following these rules, you can easily determine trigonometric values for angles on the Unit Circle. When dealing with negative angles, like \(-\frac{14\pi}{3}\), it's necessary to translate the angle into a range that's included in the Unit Circle, typically from 0 to 2π. This involves adding or subtracting multiples of 2π until you reach an equivalent angle within this range. For our example, \( t = -\frac{14\pi}{3} \) becomes \( \pi/3 \).

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the primary sin, cos, and tan functions. They include secant (sec), cosecant (csc), and cotangent (cot). Each reciprocal function is the inverse of one of the standard trigonometric functions:

• Secant: Defined as \( \text{sec}(θ) = 1/\cos(θ) \)
• Cosecant: Defined as \( \text{csc}(θ) = 1/\sin(θ) \)
• Cotangent: Defined as \( \text{cot}(θ) = 1/\tan(θ) \)

Knowing these reciprocal functions is important when you need the inverse relationship of sine, cosine, and tangent. These functions help calculate other trigonometric values and come in handy in solving more complex trigonometric equations.
Remember, the reciprocal functions are undefined if the original function's value is zero, such as \( \text{sec}(θ) \) when \( \cos(θ) = 0 \). This is because division by zero is undefined in mathematics.

Angle Simplification

Simplifying angles is crucial in trigonometry, allowing conversion of complex angles into more manageable and recognizable measures. This process involves reducing angles to an equivalent measure within a specific range, commonly 0 to 2π for radians. Here’s how you simplify the angle \(-\frac{14\pi}{3}\):

• Add multiples of 2π, which essentially wraps the angle around the unit circle into the specified range.
• Calculate and check: \(-\frac{14\pi}{3} + 2(5\pi) = \pi/3\).

This procedure brings the angle into the accessible range of the unit circle, allowing computation of trigonometric functions more conveniently.
Moreover, angle simplification aids in understanding symmetries and periodic properties of trigonometric functions, providing clearer insight into their cyclic behaviors and commonalities among different angles.