Problem 71 - Precalculus Essentials

Question

Let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c\). $$\sin (-t)-\sin t$$

Step-by-Step Solution

Answer

The expression \( \sin (-t)-\sin t \) in terms of \(a, b,\) and \(c\) is \( -2a \).

Step 1: Identify key property of sine function

The important property to use here is that sine is an odd function. This means that the sine of a negative angle is the negative of the sine of that angle. So we can write \( \sin(-t) \) as \( -\sin(t) \) or \( -a \).

Step 2: Substitute into the original expression

Now substitute \(-a\) for \(\sin(-t)\) and \(a\) for \(\sin(t)\) in the original expression \( \sin(-t)-\sin(t) \). This gets us \(-a - a = -2a\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often denoted as \( \sin \). It relates the angle of a right triangle to the length ratios of the sides. Specifically, for an angle \( t \), the sine of \( t \) is the opposite side length divided by the hypotenuse in a right triangle. This basic property helps in understanding how sine behaves for different angles.

An essential characteristic of the sine function is its periodic nature, repeating every \( 2\pi \). A significant identity about the sine function is its property as an odd function, which means \( \sin(-t) = -\sin(t) \). This property is incredibly useful in simplifying expressions involving negative angles. In the original exercise, this understanding was critical in transforming \( \sin(-t) \) into \(-a\).

Understanding these fundamental properties of the sine function is key in solving various trigonometric problems and performing transformations in trigonometric identities.

Cosine Function

The cosine function, abbreviated as \( \cos \), is another core trigonometric function. It is responsible for relating an angle in a right triangle to the adjacent side and the hypotenuse. For any angle \( t \), \( \cos(t) \) gives the length ratio of the adjacent side over the hypotenuse.

Cosine, similar to sine, is periodic with a cycle of \( 2\pi \). However, unlike sine, cosine is an even function, meaning \( \cos(-t) = \cos(t) \). This property makes cosine unique, as it reflects symmetry across the y-axis. Understanding this characteristic can be quite handy when simplifying angles in expressions.

The cosine function is crucial in various trigonometric identities and plays a vital role in transforming and simplifying trigonometric equations.

Tangent Function

The tangent function, noted as \( \tan \), is a trigonometric ratio that relates the sine and cosine functions. It is defined as the ratio of the sine function to the cosine function, meaning \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). This relationship makes tangent an important figure when angles and side ratios are involved.

One of the key features of the tangent function is that it is periodic and repeats every \( \pi \), a shorter cycle than sine and cosine. This makes tangent particularly useful in scenarios where symmetry and repetition are factors.

Tangent can exhibit undefined values when \( \cos(t) = 0 \), as division by zero does not produce a defined value. This typically occurs at odd multiples of \( \frac{\pi}{2} \). Recognizing these points is vital when working with tangent in trigonometric identities, ensuring accurate transformations and calculations.