Problem 74 - Precalculus Essentials

Question

Let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c\). $$3 \cos (-t)-\cos t$$

Step-by-Step Solution

Answer

The expression \(3 \cos (-t)-\cos t\) in terms of \(a, b,\) and \(c\) is \(2b\).

Step 1: Analyze provided trigonometric identities

We are given the equations \(\sin t=a, \cos t=b, \text { and } \tan t=c\). Also, we are asked to write the expression \(3 \cos (-t)-\cos t\) in terms of \(a, b,\) and \(c\).

Step 2: Apply the property of cosine function

The cosine function is even, which means that \(\cos (-t) = \cos t\). So, we can replace \(\cos(-t)\) with \(\cos t\). Our expression now becomes: \(3 \cos t-\cos t\).

Step 3: Simplify the expression

Combine the \(3 \cos t - \cos t\) to give \(2\cos t\). Therefore, the final expression in terms of \(t\) is \(2\cos t\).

Step 4: Replace \(\cos t\) with \(b\)

Since \(\cos t = b\), we replace \(\cos t\) with \(b\). The final expression is \(2b\).

Key Concepts

Sine and Cosine

The sine and cosine functions are fundamental in trigonometry. These functions help describe the relationship of angles within a right triangle. Whenever you encounter the sine function, it is associated with the ratio between the opposite side of the angle in a triangle and the hypotenuse. This is expressed as \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\).
Cosine, on the other hand, involves the adjacent side to the angle, making it \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).

• Both sine and cosine are crucial for understanding wave patterns, oscillations, and even physics scenarios like sound and light waves.
• They have periodic properties, meaning they repeat their values in regular intervals. For sine and cosine, this interval is every \(2\pi\).

Understanding these concepts can help simplify complex trigonometric expressions, much like in our exercise. For example, knowing that \(\cos t = b\) simplifies the conversion of any cosine-based expressions into terms of \(b\).

Tangent

The tangent function is another key player in trigonometry. It defines the relationship between the sine and cosine functions. Specifically, \(\tan \theta\) is defined as the ratio of sine to cosine, expressed simply as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
In a right-angled triangle, this is the ratio of the opposite side to the adjacent side.

• If \(\sin t = a\) and \(\cos t = b\), then \(\tan t = \frac{a}{b}\).
• Tangent can take on values from negative to positive infinity as the angle \(\theta\) approaches \((n + \frac{1}{2})\pi\), where \(n\) is any whole number. This reflects its undefined behavior at these angles where the cosine is zero.

Tangent is essential in many practical applications, including calculus and solving problems involving slopes in physics. In our specific problem set, knowing that \(\tan t = c\) allows you to easily relate additional trigonometric identities to \(c\).

Even and Odd Functions

In trigonometry, understanding even and odd functions can clarify how certain identities work. A function is even if it satisfies the condition \(f(-x) = f(x)\) for all \(x\). Conversely, a function is odd if \(f(-x) = -f(x)\).

Characteristics of Even Functions

Cosine is a quintessential even function, which leads to the identity \(\cos(-t) = \cos t\). This is a valuable property when simplifying or calculating expressions, as seen in the exercise: it allowed simplification from \(3 \cos(-t) - \cos t\) to \(2 \cos t\).

Characteristics of Odd Functions

On the contrary, the sine function is odd, meaning \(\sin(-t) = -\sin t\). This property proves helpful in solving trigonometric equations and understanding phases in waves or periodic functions.

• Recognizing whether a function is even or odd swiftly reduces the complexity of problem-solving in trigonometry.

By identifying these properties, problems become more manageable, enabling simplification without getting entangled in complex calculations.