Problem 1 - Calculus Single Variable

Question

Calculate each of the six trigonometric functions at angle \(\theta\) without using a calculator. \(\theta=\pi / 6\)

Step-by-Step Solution

Answer

\(\sin(\pi/6) = \frac{1}{2}\), \(\cos(\pi/6) = \frac{\sqrt{3}}{2}\), \(\tan(\pi/6) = \frac{\sqrt{3}}{3}\), \(\csc(\pi/6) = 2\), \(\sec(\pi/6) = \frac{2\sqrt{3}}{3}\), \(\cot(\pi/6) = \sqrt{3}\).

Step 1: Identify Reference Angle

The angle \( \theta = \pi / 6 \) is a known angle in trigonometry. In degrees, this angle is equivalent to \( 30^\circ \).

Step 2: Use Known Values from Unit Circle

For \( \theta = \pi / 6 \), the corresponding point on the unit circle is \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \). These values represent \( \cos(\theta) \) and \( \sin(\theta) \), respectively.

Step 3: Calculate Sine and Cosine

From the unit circle, we have \( \sin(\pi / 6) = \frac{1}{2} \) and \( \cos(\pi / 6) = \frac{\sqrt{3}}{2} \).

Step 4: Calculate Tangent

The tangent function is defined as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Substituting the values, \( \tan(\pi / 6) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \). Simplifying gives \( \tan(\pi / 6) = \frac{\sqrt{3}}{3} \) after rationalizing the denominator.

Step 5: Calculate Cosecant

The cosecant is the reciprocal of sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Thus, \( \csc(\pi / 6) = \frac{1}{\frac{1}{2}} = 2 \).

Step 6: Calculate Secant

The secant is the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Thus, \( \sec(\pi / 6) = \frac{1}{\frac{\sqrt{3}}{2}} \), which simplifies to \( \frac{2}{\sqrt{3}} \). Rationalizing gives \( \frac{2\sqrt{3}}{3} \).

Step 7: Calculate Cotangent

The cotangent is the reciprocal of tangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} \). So, \( \cot(\pi / 6) = \frac{\sqrt{3}}{1} = \sqrt{3} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, essential for understanding the calculation of trigonometric functions. It refers to a circle with a radius of one, centered at the origin (0, 0) of a coordinate plane. The importance of the unit circle lies in its ability to define the trigonometric functions for all real angles.

The unit circle is often represented graphically, where the angle in question is measured from the positive x-axis within a counter-clockwise direction. When we talk about angles like \( \theta = \pi / 6 \) or \( 30^\circ \), it helps to visualize where this angle is positioned on the circle.

• Point Representation: Each angle on the unit circle corresponds to a point \((x, y)\) where \(x = \cos(\theta)\) and \(y = \sin(\theta)\).
• Degrees vs Radians: Angle measurements can be in degrees or radians, and \( \pi / 6 \) radians is equivalent to \(30^\circ\).

Visualizing the unit circle and knowing how to identify these points for common angles provides a powerful tool for calculating the trigonometric functions easily. It helps in converting spatialrelationships into algebraic expressions which we use in trigonometry.

Sine and Cosine Calculation

In trigonometry, the sine and cosine functions are among the most recognized and utilized. Understanding their calculation is crucial when working with angles, especially ones located on the unit circle. For \( \theta = \pi / 6 \), the points \( ( \frac{\sqrt{3}}{2}, \frac{1}{2} ) \) on the unit circle directly inform us of \( \cos(\pi / 6) \) and \( \sin(\pi / 6) \) respectively.

The relationships are straightforward:

• Sine: The y-coordinate of the unit circle at any angle \(\theta\) represents the sine value. Thus, \( \sin(\pi / 6) = \frac{1}{2} \).
• Cosine: The x-coordinate indicates the cosine value, giving us \( \cos(\pi / 6) = \frac{\sqrt{3}}{2}\).

To leverage these values, focus on learning the common angles on the unit circle, such as \( 0, \pi / 6, \pi / 4, \pi / 3, \text{and} \pi / 2 \). Having this knowledge allows you to calculate not only sine and cosine values manually but also helps in solving for other trigonometric functions by using identities and relationships among them.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions: sine, cosine, and tangent. These include cosecant, secant, and cotangent, and are essential for understanding more complex trigonometric problems.

Here's how they relate:

• Cosecant (\(\csc\)): It is the reciprocal of sine. For \(\theta = \pi / 6\), \(\csc(\pi / 6) = \frac{1}{\sin(\pi / 6)} = 2\).
• Secant (\(\sec\)): It is the reciprocal of cosine. Thus, \(\sec(\pi / 6) = \frac{1}{\cos(\pi / 6)}\), simplifying to \(\frac{2\sqrt{3}}{3}\) after rationalization.
• Cotangent (\(\cot\)): It is the reciprocal of tangent. Computed as \(\cot(\pi / 6) = \frac{1}{\tan(\pi / 6)} = \sqrt{3}\).

Using reciprocal relationships allows for deeper insights into trigonometric computations. Each reciprocal function gives a different perspective, expanding the ability to calculate angles and sides in various contexts. Mastering these reciprocal functions provides a solid foundation for progressing in trigonometry.