Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. $$\tan ^{-1}(\tan (3 \pi / 4))$$

Short answer

In the expression $$\tan^{-1}(\tan(3\pi/4))$$, we first found the angle inside the tangent function, which is \(3\pi/4\) radians. Then, we evaluated the tangent function for that angle, resulting in a value of $$-1$$. Finally, we applied the inverse tangent function to find the angle whose tangent is equal to -1, which is $$-\frac{\pi}{4}$$. Therefore, $$\tan^{-1}(\tan(3\pi/4)) = -\frac{\pi}{4}$$.

Step-by-step solution

Find the angle inside the tangent function

The first step is to find the angle inside the tangent function which is \(3\pi/4\) radians.

Evaluate the tangent function

Next, we need to evaluate the tangent function at the angle \(3\pi/4\). Recall the definition of the tangent function as the ratio of the sine and cosine of the angle, i.e., $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$ Applying this to our angle, we have: $$\tan(3\pi/4) = \frac{\sin(3\pi/4)}{\cos(3\pi/4)}$$ Notice that \(3\pi/4\) is in the second quadrant, where sine is positive and cosine is negative. Looking at the values for sine and cosine at the standard angles (\(\pi/4, \pi/2, \pi\)), we can use symmetry and find the sine and cosine values for \(3\pi/4\): $$\sin(3\pi/4) = \sin(\pi - \pi/4) = \sin(\pi/4) = \frac{\sqrt{2}}{2}$$ $$\cos(3\pi/4) = -\cos(\pi/4) = -\frac{\sqrt{2}}{2}$$ Now we can plug these values into our tangent function: $$\tan(3\pi/4) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1$$

Apply the inverse tangent function

Now we need to find the angle whose tangent is equal to -1. We know the inverse tangent function has a range from \(-\pi/2\) to \(\pi/2\). Knowing that \(\tan(\theta)=-1\), we can solve for the angle: $$\tan^{-1}(-1) = -\frac{\pi}{4}$$ Finally, we can plug this value back into the original expression: $$\tan^{-1}(\tan(3\pi/4)) = \tan^{-1}(-1) = -\frac{\pi}{4}$$

Key concepts

Evaluating Trigonometric Functions

Understanding how to evaluate trigonometric functions is essential in fields like mathematics, physics, and engineering. Trigonometric functions relate the angles of a triangle to its side lengths and can be extended to describe rotations and periodic phenomena.

When evaluating trigonometric functions, it's crucial to remember their definitions and the properties of angles in different quadrants. For instance, the sign of the trigonometric functions changes depending on the quadrant the angle falls in. In our initial problem, evaluating the tangent function at the angle \(3\pi/4\) required understanding that this angle is located in the second quadrant, where sine is positive and cosine is negative.

Evaluating trigonometric functions often involves identifying symmetrical properties and using reference angles. A reference angle is the acute angle that a standard position angle makes with the x-axis, and it allows us to easily determine the value of trigonometric functions for more complex angles by comparing them to the corresponding acute angle in the first quadrant.

It's also important to keep in mind that evaluating an inverse trigonometric function means finding the angle that, when the trigonometric function is applied, gives you the original value. This is done within the restricted range of the inverse function to ensure that each value corresponds to exactly one angle.

Tangent Function

The tangent function, represented by \(\tan(\theta)\), is one of the fundamental trigonometric functions. The value of the tangent function can be understood as the ratio of the opposite side to the adjacent side in a right-angled triangle, or more generally, as the ratio of the sine function to the cosine function:

\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]

In the context of our exercise, we evaluated the tangent of the angle \(3\pi/4\) by using this ratio. Since \(3\pi/4\) is an angle in the second quadrant, the sign of each function—sine positive and cosine negative—affects the outcome, resulting in a negative value. It is essential to recognize that the tangent function is periodic with a period of \(\pi\), which means it repeats its values every \(\pi\) radians and is undefined when the cosine function is zero, like at \(\pi/2\) and \(3\pi/2\).

When working with the tangent function, be mindful of its graph and properties. The graph shows a repeating pattern with vertical asymptotes, highlighting the points where the function is undefined. Understanding these characteristics can help solve problems involving the tangent function more effectively.

Radian Measure

Radian measure is a way of expressing angles based on the radius of a circle. It's a fundamental concept in trigonometry and calculus that provides a clear connection between linear and angular measurements.

One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. The circumference of a circle is \(2\pi r\), and since the circumference is the arc corresponding to the full angle around the circle, there are \(2\pi\) radians in a complete revolution.

In our example, we dealt with an angle measured in radians: \(3\pi/4\). Understanding radians is crucial for evaluating trigonometric functions at these angles. It lets us use the properties of the unit circle and the symmetry of trigonometric functions to accurately find function values without necessarily relying on memorization.

Radian measure is often preferred over degrees in higher-level mathematics because it simplifies many formulas and processes, such as differentiation and integration of trigonometric functions. Conversions between degrees and radians are also important to master, as different contexts may call for different units of angle measurement. Keep in mind that \(\pi\) radians equals 180 degrees, which allows you to switch between these units when necessary.