Problem 119 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the value of the trigonometric function to evaluate each function. \(\cos (-t)=-\frac{1}{5}\) (a) \(\cos t\) (b) \(\sec (-t)\)

Step-by-Step Solution

Answer

The solution for (a) \( \cos t \) is -1/5, and for (b) \( \sec (-t) \) is -5.

Step 1: Evaluate \( \cos t \)

Because cosine function is even, we can write \( \cos(-t)=\cos(t) \) . So, \( \cos t=-1/5 \).

Step 2: Evaluate \( \sec (-t) \)

Now, secant function is the reciprocal of the cosine function, which means \( \sec(t)=1/\cos(t) \). As we have \( \cos(-t)=-1/5 \), therefore \( \sec(-t)=1/\cos(-t) = -5 \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often abbreviated as "cos." It relates an angle in a right triangle to the ratio of the adjacent side to the hypotenuse. In the unit circle, the cosine of an angle gives the horizontal coordinate of the corresponding point on the circle. Mathematically, it is expressed as:\[\cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\]The cosine function is periodic and even, meaning it repeats its values in a predictable manner as the angle increases. This periodicity is one of its critical features. The cosine function's period is \(2\pi\), meaning it repeats every \(360^\circ\).
In terms of its graph, the cosine function starts at 1 for an angle of 0 degrees, decreases to -1 at \(180^\circ\), and goes back to 1 at \(360^\circ\). This wave-like shape is symmetric about the vertical axis, showing its even nature.

Even Function

An even function is a function that satisfies the condition \(f(-x) = f(x)\) for every value of \(x\) in its domain. This means the function's graph is symmetric with respect to the vertical axis. Most importantly, for trigonometric functions like cosine, this property makes calculations more straightforward.
Certain common properties of even functions include:- Symmetrical graph: The left side of the graph is a mirror image of the right.- Examples of even functions include \(x^2\), \(\cos(x)\) and constants.- For even functions, knowing the function's value at negative inputs gives us the value at positive but opposite inputs instantly.Because \(\cos(-t) = \cos(t)\), when given \(\cos(-t) = -\frac{1}{5}\), we can directly conclude that \(\cos t = -\frac{1}{5}\) without additional computation. This reduces the need for more complex algebraic manipulation when dealing with trigonometric identities.

Secant Function

The secant function, notated as \(\sec\), is the reciprocal of the cosine function. Mathematically, it is written as:\[\sec(\theta) = \frac{1}{\cos(\theta)}\]It plays an essential role in trigonometry, especially when evaluating angles where the cosine is non-zero.
Key points about the secant function include:- Because the cosine function can never equal zero, the secant function will be undefined wherever \(\cos(\theta) = 0\).- The graph of \(\sec\) resembles that of \(\cos\) but flipped and stretched, maintaining infinity at the points where cosine is zero.
Given \(\cos(-t) = -\frac{1}{5}\), as in the exercise, the secant can be easily found by flipping the value of cosine. Thus, \(\sec(-t) = \frac{1}{-\frac{1}{5}} = -5\). This shows how understanding the reciprocal nature of secant simplifies calculations.