Problem 35 - Algebra and Trigonometry

Question

In Exercises \(33-42,\) let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c .\) $$ 4 \cos (-t)-\cos t $$

Step-by-Step Solution

Answer

The simplified form of the expression \(4 \cos (-t)-\cos t\) in terms of \(a\), \(b\), and \(c\) is \(3b\).

Step 1: Apply the Property of Cosine Function to the Expression

Replacing \(\cos(-t)\) with \(\cos(t)\) in the expression, using the property that cosine is an even function, the expression \(4 \cos (-t)-\cos t\) becomes \(4 \cos t - \cos t\).

Step 2: Simplify the Expression

To simplify the expression, combine the like terms. The expression \(4 \cos (t) - \cos t\) simplifies to \(3 \cos(t)\).

Step 3: Substitute the Value of \(\cos(t)\)

We know that \(\cos(t)\) equals \(b\), using this information, replace \(\cos(t)\) with \(b\) in the simplified expression. The expression \(3 \cos(t)\) becomes \(3b\).

Key Concepts

Cosine Function

When tackling trigonometry problems, a solid understanding of the cosine function plays a crucial role. It is one of the primary functions in trigonometry and relates to the coordinate of a point on the unit circle. Specifically, if we consider a right-angled triangle with an angle \(t\), the cosine of this angle, \(\cos(t)\), is defined as the ratio of the adjacent side to the hypotenuse.

In many trigonometric applications, knowing the exact value is essential, but the cosine function also has special characteristics. For example, it is an even function, meaning that \(\cos(t) = \cos(-t)\). This is particularly useful when simplifying expressions involving cosine, as seen in the provided exercise. Recognizing such properties allows for quick simplification of expressions and understanding relations between different trigonometric functions, as the cosine function is directly related to the sine function via the Pythagorean identity: \(\cos^2(t) + \sin^2(t) = 1\).

Even and Odd Trigonometric Functions

Trigonometric functions can be classified as even or odd, which is a reference to their symmetry properties. An even function is symmetric about the y-axis, meaning the function's value at positive \(t\) equals its value at negative \(t\). The cosine function, \(\cos(t)\), is an even function because \(\cos(t) = \cos(-t)\).

On the other hand, an odd function has rotational symmetry about the origin, which means that \(f(-t) = -f(t)\). The sine and tangent functions are examples of odd functions since \(\sin(-t) = -\sin(t)\) and \(\tan(-t) = -\tan(t)\). Knowing whether a trigonometric function is even or odd is invaluable — it can drastically simplify an expression since you can replace the function of a negative argument with the function of a positive one or its negative, depending on the property.

Trigonometric Expression Simplification

Simplifying trigonometric expressions can often feel like solving a puzzle. The process involves several key steps, such as recognizing properties of trigonometric functions, combining like terms, and substituting known values. In the case of the exercise given, simplification began with understanding the even property of the cosine function. This allowed for combining the terms \(4 \cos (-t)\) and \(-\cos t\) into \(4 \cos t - \cos t\), a clear instance of combining like terms.

Once like terms are combined, any constants or known values can be substituted to express the result in a more straightforward and usable form. Here, knowing that \(\cos(t)\) is represented by \(b\), let us turn the trigonometric expression into the algebraic expression \(3b\). Recognizing these patterns and applying the simplification steps can reduce complex trigonometric expressions to their essence, making them much easier to work with.