Problem 120 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the value of the trigonometric function to evaluate each function. \(\sin (-t)=\frac{3}{8}\) (a) \(\sin t\) (b) \(\csc t\)

Step-by-Step Solution

Answer

The value for \( \sin t \) is -3/8 and for \( \csc t \) it is -8/3.

Step 1: Find the value of \(\sin t\)

Given that the value of \( \sin(-t) = \frac{3}{8} \), and the property of sine for negative angles, we know that \( \sin(-t) = - \sin t \). Therefore, \( \sin t = -\frac{3}{8} \).

Step 2: Calculate \(\csc t\)

The cosecant function, \( \csc t \), is the reciprocal of the sine function. Hence, \( \csc t = \frac{1}{\sin t} \). Substitute the value from Step 1 to get \( \csc t = -\frac{8}{3} \).

Key Concepts

Understanding Sine: The Fundamental Trigonometric Function

The sine function, represented as \( \sin \), is one of the basic trigonometric functions you'll encounter in mathematics. It has a vital role in understanding periodic phenomena such as waves and circular motion. In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Beyond triangles, it extends to the unit circle, where it's defined as the \( y \)-coordinate of a point on the circle.

Key points about sine:

• The sine function can take any real number as its input, which is often expressed in degrees or radians.
• Sine values range from -1 to 1.
• \( \sin(0) = 0 \), \( \sin(\pi/2) = 1 \), and \( \sin(\pi) = 0 \).

Understanding these properties helps in solving problems involving periodic behaviors. It's crucial to become familiar with this function before delving deeper into other trigonometric concepts.

Cosecant: The Reciprocal of Sine

The cosecant function, \( \csc \), might seem unfamiliar initially but is simply the reciprocal of the sine function. Where sine is naturally bounded between -1 and 1, the cosecant function shows us those peaks and troughs as values outside this range, emphasizing its nature as a reciprocal.

Key aspects of cosecant include:

• It is defined as \( \csc t = \frac{1}{\sin t} \).
• Cosecant is undefined wherever sine is zero, such as at \( t = n\pi \), where \( n \) is an integer, because division by zero is not allowed.
• When evaluating cosecant, ensure the sine value is known first, as this will directly determine the cosecant value.

In trigonometric calculations, getting the sine value first often helps in efficiently finding cosecant, just like we did in calculating \( \csc t = -\frac{8}{3} \) using \( \sin t = -\frac{3}{8} \).

Handling Negative Angles in Trigonometry

In trigonometry, negative angles introduce an interesting twist to function values. They occur typically in a clockwise direction from the initial line on a coordinate plane, opposite to the conventional counter-clockwise positive angle measurement.

Regarding their impact on trigonometric functions, an essential property of sine is that \( \sin(-t) = -\sin t \). It indicates that sine is an odd function, which reflects through the origin, resulting in the negation of sine values when altering the angle.

• The property \( \sin(-t) = -\sin t \) was essential in solving for \( \sin t \) when given \( \sin(-t) = \frac{3}{8} \).
• Such manipulation aids in understanding how angles and their trigonometric responses relate spatially within the coordinate system.

Grasping how negative angles influence function values is crucial in higher-level mathematics, physics, and engineering applications.