Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function for the given values of \(t\). Values of \(t\) (a) \(t=3\) (a) \(t=2\) (a) \(t=0.3\) (a) \(t=0.1\) (b) \(t=\pi\) (b) \(t=0\) (b) \(t=-\pi / 4\) (b) \(t=-2 \pi / 3\) Trigonometric Function $$f(t)=\tan t$$

Step-by-Step Solution

Answer

The solutions for the tangent of the given values are: (rounding to 2 decimal places) For \(t=3, f(3) = -0.14\). For \(t=2, f(2) = -2.19\). For \(t=0.3, f(0.3) = 0.31\). For \(t=0.1, f(0.1) = 0.10\). For \(t=π, f(π) = 0.00\). For \(t=0, f(0) = 0.00\). For \(t=-π / 4, f(-π / 4) = -1.00\). For \(t=-2π / 3, f(-2π / 3) = 0.58\)

Step 1: Substitute \(t\) by the given values (Part 1)

Substitute \(t\) in \(f(t) = \tan t\) by the first 4 given values and evaluate the function: \n For (a1) \(t=3, f(3) = \tan(3)\) \n For (a2) \(t=2, f(2) = \tan(2)\) \n For (a3) \(t=0.3, f(0.3) = \tan(0.3)\) \n For (a4) \(t=0.1, f(0.1) = \tan(0.1)\)

Step 2: Substitute \(t\) by the given values (Part 2)

Next, substitute \(t\) by the last 4 given values and evaluate the function: \n For (b1) \(t=π, f(π) = \tan(π)\) \n For (b2) \(t=0, f(0) = \tan(0)\) \n For (b3) \(t=-π / 4, f(-π / 4) = \tan(-π / 4)\) \n For (b4) \(t=-2π / 3, f(-2π / 3) = \tan(-2π / 3)\)

Step 3: Transform Answers to Decimal Form

Finally, using a calculator, we get the numerical values of the function for the different input values, rounded at 2 decimal places: \n For \(t=3, f(3) = -0.14\) \n For \(t=2, f(2) = -2.19\) \n For \(t=0.3, f(0.3) = 0.31\) \n For \(t=0.1, f(0.1) = 0.10\) \n For \(t=π, f(π) = 0.00\) \n For \(t=0, f(0) = 0.00\) \n For \(t=-π / 4, f(-π / 4) = -1.00\) \n For \(t=-2π / 3, f(-2π / 3) = 0.58\)

Key Concepts

Understanding the Tangent Function

The tangent function, represented as \( \tan(t) \), is a fundamental trigonometric function that relates the angle \( t \) to the ratio of the opposite side to the adjacent side in a right triangle. It is one of the three primary trigonometric functions, with sine and cosine being the other two. The tangent function can take any real number as its input, unlike sine and cosine, which are restricted within specific ranges.The tangent function is periodic, which means it repeats its values in regular intervals of \( \pi \) radians. This means every \( \pi \) radians, the function returns to the same point. This periodicity can help calculate values without a calculator if you know the unit circle's key points.However, it's important to remember that the tangent function is undefined at certain points, specifically where the cosine is zero, such as at \( \pi/2 \), \( 3\pi/2 \), and so on. This is because you cannot divide by zero, and at these points, the tangent's formula results in a division by zero scenario. Exploring these zero and undefined points helps in understanding the graph of the tangent, which has a distinct repeated vertical asymptote at these locations.

Steps in Evaluating Functions

Evaluating a trigonometric function, like the tangent, involves substituting a given value for \( t \) into the function. Here, you would use the function \( f(t) = \tan t \). The process starts by replacing \( t \) with a specific number and finding the value of the function at that point.For example:- When \( t = 3 \), you substitute this into the tangent function, resulting in \( \tan(3) \). - Similarly, for \( t = \pi \), you replace \( t \) with \( \pi \) in the function to get \( \tan(\pi) \). After substituting the values, use a calculator or a trigonometric table to find the tangent's measurement. It’s especially important to ensure that any angles measured are in the correct unit, either radians or degrees, as calculating with the wrong unit will yield incorrect results.Evaluating these functions helps in understanding the behavior of tangent at various points, showing how it rises, falls, repeats, and becomes undefined.

Finding Decimal Approximations

Once you have substituted your values into the tangent function, the next step is to get the decimal approximation of the result. Since the tangent of any angle that isn't a key unit circle point can result in an irrational number, calculators are often used to find an approximate decimal value.Decimal approximations facilitate easier readability and understanding, especially in practical applications, where precise values may be unnecessary. For most purposes, rounding these decimals to two decimal places suffices. This way:- For \( t = 3 \), \( \tan(3) \approx -0.14 \).- Similarly, for \( t = -\pi/4 \), \( \tan(-\pi/4) \approx -1.00 \).Finding and using decimal approximations allows you to quickly grasp the scale and proportions of tangent values, essential in fields like engineering and physics, where rapid calculations of angles frequently occur. A clear understanding of rounding conventions and when an approximation is appropriate can greatly aid in producing accurate, functional work.