Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate without using a calculator. (a) \(\tan \frac{\pi}{6}\) (b) \(\sec \pi\) (c) \(\sec \frac{3 \pi}{4}\) (d) \(\csc \frac{\pi}{2}\) (e) \(\cot \frac{\pi}{4}\) (f) \(\tan \left(-\frac{\pi}{4}\right)\)

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{3}}{3}\), (b) \(-1\), (c) \(-\sqrt{2}\), (d) \(1\), (e) \(1\), (f) \(-1\)

Step 1: Evaluate \(\tan \frac{\pi}{6}\)

We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). For \(\theta = \frac{\pi}{6}\), \(\sin \frac{\pi}{6} = \frac{1}{2}\) and \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\). Thus, \(\tan \frac{\pi}{6} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).

Step 2: Evaluate \(\sec \pi\)

The secant function is the reciprocal of the cosine function, \(\sec \theta = \frac{1}{\cos \theta}\). For \(\theta = \pi\), \(\cos \pi = -1\). Therefore, \(\sec \pi = \frac{1}{-1} = -1\).

Step 3: Evaluate \(\sec \frac{3\pi}{4}\)

For \(\theta = \frac{3\pi}{4}\), \(\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\). Thus, \(\sec \frac{3\pi}{4} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\).

Step 4: Evaluate \(\csc \frac{\pi}{2}\)

The cosecant function is the reciprocal of the sine function, \(\csc \theta = \frac{1}{\sin \theta}\). For \(\theta = \frac{\pi}{2}\), \(\sin \frac{\pi}{2} = 1\). Hence, \(\csc \frac{\pi}{2} = \frac{1}{1} = 1\).

Step 5: Evaluate \(\cot \frac{\pi}{4}\)

The cotangent function is the reciprocal of the tangent function, \(\cot \theta = \frac{1}{\tan \theta}\). For \(\theta = \frac{\pi}{4}\), \(\tan \frac{\pi}{4} = 1\). Therefore, \(\cot \frac{\pi}{4} = \frac{1}{1} = 1\).

Step 6: Evaluate \(\tan -\frac{\pi}{4}\)

The tangent function is an odd function, meaning \(\tan(-\theta) = -\tan(\theta)\). Thus, \(\tan -\frac{\pi}{4} = -\tan \frac{\pi}{4} = -1\).

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function denoted as \(\tan \theta\). It is defined as the ratio of the sine and cosine functions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). This simple relationship makes it easy to find the tangent of an angle if you already know its sine and cosine values. For example, to find \(\tan \frac{\pi}{6}\), we use the values \(\sin \frac{\pi}{6} = \frac{1}{2}\) and \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\). Plugging these into the formula gives \(\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Another important property of the tangent function is that it is an odd function. This means that \(\tan(-\theta) = -\tan(\theta)\). Such properties are useful when dealing with negative angles, as seen in \(\tan(-\frac{\pi}{4}) = -1\).
Overall, the tangent function helps in understanding the relationship between the legs of a right triangle and the angle, providing critical insights into trigonometry.

Secant Function

The secant function is another essential trigonometric function, represented as \(\sec \theta\). It is defined as the reciprocal of the cosine function: \(\sec \theta = \frac{1}{\cos \theta}\). Knowing the cosine of an angle allows us to compute its secant easily.
For example:

• To find \(\sec \pi\), we use \(\cos \pi = -1\), giving us \(\sec \pi = \frac{1}{-1} = -1\).
• To compute \(\sec \frac{3\pi}{4}\), knowing \(\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\) leads to \(\sec \frac{3\pi}{4} = -\sqrt{2}\).

The secant function is vital because it extends our understanding of the cosine function, especially when analyzing angles in various quadrants of the unit circle. Sometimes, secant values give a clearer view of an angle's property, illuminating aspects of trigonometric identity relationships.

Cosecant Function

The cosecant function, denoted as \(\csc \theta\), is the reciprocal of the sine function. Its formula is \(\csc \theta = \frac{1}{\sin \theta}\). This function plays an important role when we need the relationship between an angle and its sine expressed inversely.
For instance:

• To find \(\csc \frac{\pi}{2}\), note that \(\sin \frac{\pi}{2} = 1\), which makes \(\csc \frac{\pi}{2} = \frac{1}{1} = 1\).

The cosecant function is particularly useful when evaluating angles that result in zero for the sine function. In such cases, \(\csc \theta\) is undefined because the reciprocal of zero is undefined. Understanding the cosecant function enriches your grasp of trigonometric properties and helps in solving complex identities.

Cotangent Function

The cotangent function is symbolized by \(\cot \theta\) and is defined as the reciprocal of the tangent function: \(\cot \theta = \frac{1}{\tan \theta}\). It can also be expressed as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), offering an alternative perspective on the tangent relationship.
Here's a concise example:

• For \(\cot \frac{\pi}{4}\), since \(\tan \frac{\pi}{4} = 1\), it follows that \(\cot \frac{\pi}{4} = \frac{1}{1} = 1\).

The cotangent function is extremely useful in various trigonometric applications, particularly when analyzing relationships involving angles in right triangles. Since the cotangent meets the tangent at critical points, it can help simplify complex expressions and deepen your understanding of trigonometric identities.