Problem 3 - Calculus with Concepts in Calculus

Question

Find the following values. a. \(\sin \frac{11 \pi}{6}\) b. \(\sin \left(-\frac{2 \pi}{3}\right)\) c. \(\cos \frac{5 \pi}{4}\) d. \(\cos \left(-\frac{7 \pi}{6}\right)\) e. \(\tan \frac{4 \pi}{3}\) f. \(\tan \left(-\frac{\pi}{4}\right)\) g. \(\cot \frac{\pi}{6}\) h. \(\cot \left(-\frac{17 \pi}{3}\right) \quad \mathbf{i} . \sec 3 \pi\) j. \(\sec \left(-\frac{\pi}{3}\right)\) k. \(\csc \frac{\pi}{2}\) L. \(\csc \left(-\frac{5 \pi}{3}\right)\)

Step-by-Step Solution

Answer

a. \(-\frac{1}{2}\), b. \(-\frac{\sqrt{3}}{2}\), c. \(-\frac{\sqrt{2}}{2}\), d. \(-\frac{\sqrt{3}}{2}\), e. \(\sqrt{3}\), f. \(-1\), g. \(\sqrt{3}\), h. \(-\frac{1}{\sqrt{3}}\), i. \(-1\), j. \(2\), k. \(1\), L. \(\frac{2}{\sqrt{3}}\).

Step 1: Find \(\sin \frac{11 \pi}{6}\)

Convert \(\frac{11\pi}{6}\) into degrees by multiplying by \(\frac{180}{\pi}\) to get \(\frac{11\pi}{6} \times \frac{180}{\pi} = 330^\circ\). The sine of \(330^\circ\) is the same as \(-\sin 30^\circ\), due to it being in the fourth quadrant where sine is negative. Thus, \(\sin \frac{11\pi}{6} = -\frac{1}{2}\).

Step 2: Calculate \(\sin \left(-\frac{2 \pi}{3}\right)\)

Convert \(-\frac{2\pi}{3}\) to degrees to get \(-\frac{2\pi}{3} \times \frac{180}{\pi} = -120^\circ\). Sine is an odd function, so \(\sin(-x) = -\sin(x)\). Thus, \(\sin\left(-120^\circ\right) = -\sin(120^\circ)\). Since \(\sin(120^\circ) = \frac{\sqrt{3}}{2}\), it follows that \(\sin\left(-\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{2}\).

Step 3: Determine \(\cos \frac{5 \pi}{4}\)

Convert \(\frac{5\pi}{4}\) to degrees: \(\frac{5\pi}{4} \times \frac{180}{\pi} = 225^\circ\). Since \(225^\circ\) is in the third quadrant, where cosine is negative and equivalent to \(\cos 45^\circ\), \(\cos 225^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2}\).

Step 4: Find \(\cos \left(-\frac{7 \pi}{6}\right)\)

Convert \(-\frac{7\pi}{6}\) to degrees to get \(-\frac{7\pi}{6} \times \frac{180}{\pi} = -210^\circ\). By the even property of cosine, \(\cos(-x) = \cos(x)\), so \(\cos\left(-210^\circ\right) = \cos(210^\circ)\). Since \(\cos(210^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}\), \(\cos \left(-\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\).

Step 5: Calculate \(\tan \frac{4 \pi}{3}\)

Convert \(\frac{4\pi}{3}\) to degrees: \(\frac{4\pi}{3} \times \frac{180}{\pi} = 240^\circ\). In the third quadrant, where tangent is positive, \(\tan 240^\circ = \tan 60^\circ = \sqrt{3}\). Therefore, \(\tan \frac{4\pi}{3} = \sqrt{3}\).

Step 6: Determine \(\tan \left(-\frac{\pi}{4}\right)\)

Convert \(-\frac{\pi}{4}\) to degrees: \(-\frac{\pi}{4} \times \frac{180}{\pi} = -45^\circ\). Tangent is an odd function, so \(\tan(-x) = -\tan(x)\). Therefore, \(\tan\left(-45^\circ\right) = -\tan(45^\circ) = -1\), so \(\tan \left(-\frac{\pi}{4}\right) = -1\).

Step 7: Find \(\cot \frac{\pi}{6}\)

Given \(\cot(x) = \frac{1}{\tan(x)}\), find \(\tan \frac{\pi}{6}\), which is \(\frac{1}{\sqrt{3}}\). Thus, \(\cot \frac{\pi}{6} = \sqrt{3}\).

Step 8: Calculate \(\cot \left(-\frac{17 \pi}{3}\right)\)

First, reduce \(-\frac{17\pi}{3}\) to an equivalent angle between \(-2\pi\) and \(2\pi\) by adding \(6\pi\) to get \(\frac{\pi}{3}\). Thus, \(\cot \left(-\frac{17 \pi}{3}\right) = \cot \left(-\frac{\pi}{3}\right)\). Since \(\cot(x) = \frac{1}{\tan(x)}\), and \(\tan \left(-\frac{\pi}{3}\right) = -\sqrt{3}\), \(\cot \left(-\frac{\pi}{3}\right) = -\frac{1}{\sqrt{3}}\).

Step 9: Determine \(\sec 3 \pi\)

Convert \(3\pi\) to degrees: \(3\pi \times \frac{180}{\pi} = 540^\circ\). Since \(540^\circ\) is a full circle plus \(180^\circ\), \(\cos 540^\circ = -1\), hence \(\sec 3\pi = \frac{1}{\cos 540^\circ} = -1\).

Step 10: Calculate \(\sec \left(-\frac{\pi}{3}\right)\)

Convert \(-\frac{\pi}{3}\) to degrees: \(-\frac{\pi}{3} \times \frac{180}{\pi} = -60^\circ\). Since \(\cos(-x) = \cos(x)\), \(\cos(-60^\circ) = \cos(60^\circ) = \frac{1}{2}\). Hence \(\sec \left(-\frac{\pi}{3}\right) = \frac{1}{\cos \left(-\frac{\pi}{3}\right)} = 2\).

Step 11: Find \(\csc \frac{\pi}{2}\)

Since \(\sin \frac{\pi}{2} = 1\), \(\csc \frac{\pi}{2} = \frac{1}{\sin \frac{\pi}{2}} = 1\).

Step 12: Determine \(\csc \left(-\frac{5 \pi}{3}\right)\)

Convert \(-\frac{5\pi}{3}\) to an angle in standard position: \(\frac{5\pi}{3} \approx -300^\circ\), which is \(60^\circ\) in positive angle. Since \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), \(\csc \left(-\frac{5\pi}{3}\right) = \frac{1}{\sin \left(-\frac{5 \pi}{3}\right)} = \frac{2}{\sqrt{3}}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, especially in relation to circles and periodic phenomena. They include the six key functions: sine, cosine, tangent, cotangent, secant, and cosecant. Each of these functions relates to the angles and sides of a right triangle, serving as a foundation in trigonometric studies.

• Sine ( \(\sin\)): opposite side over hypotenuse.
• Cosine ( \(\cos\)): adjacent side over hypotenuse.
• Tangent ( \(\tan\)): opposite side over adjacent side.
• Cotangent ( \(\cot\)): reciprocal of tangent.
• Secant ( \(\sec\)): reciprocal of cosine.
• Cosecant ( \(\csc\)): reciprocal of sine.

Understanding these functions and how they relate to different angles and circles helps in solving various mathematical problems.

Angle Conversion

Angle conversion is a crucial step in solving trigonometric problems, as it often requires converting radians to degrees or vice versa. This allows for a more intuitive understanding of angles for calculation.

To convert from radians to degrees, you multiply by 180 and divide by \(\pi\). For example, converting \(\frac{11\pi}{6}\) gives \(\frac{11\pi}{6} \times \frac{180}{\pi} = 330^\circ\). Similarly, converting negative angles, like \(-\frac{2\pi}{3}\), involves the same process: \(-\frac{2\pi}{3} \times \frac{180}{\pi} = -120^\circ\).

Understanding this conversion helps to determine the function's sign and quadrant positioning.

Unit Circle

The unit circle is a powerful tool in trigonometry, displaying angles and the corresponding sine, cosine, and tangent values. It is a circle with a radius of one, centered at the origin of a coordinate system. On the unit circle:

• The x-coordinate represents \(\cos\theta\).
• The y-coordinate represents \(\sin\theta\).

Angles originate from the positive x-axis, measured counterclockwise for positive angles and clockwise for negative angles. The unit circle also illustrates the cyclical nature of trigonometric functions, explaining why functions repeat values at specific intervals.

For example, locating \(\frac{5\pi}{4}\) on the unit circle shows it in the third quadrant, where sine is negative. The unit circle thus clarifies how quadrants affect the function signs.

Sine

Sine, represented as \(\sin\theta\), is one of the fundamental trigonometric functions. It correlates the angle \(\theta\) to the y-coordinate on the unit circle. Sine values range between -1 and 1, varying based on the angle's position and quadrant.

Sine has key properties:

• An odd function: \(\sin(-x) = -\sin(x)\).
• Negative in the third and fourth quadrants.

For instance, in solving \(\sin \frac{11\pi}{6}\), we find it to be \(-\frac{1}{2}\). Observing angle mapping helps determine the sine function's behavior, ensuring accurate trigonometric outcomes.

Cosine

Cosine, indicated as \(\cos\theta\), is another core trigonometric function related to the x-coordinate on the unit circle. Like sine, cosine values range between -1 and 1, influenced by the angle's position relative to the quadrants.

Key properties of cosine include:

• An even function: \(\cos(-x) = \cos(x)\).
• Negative in the second and third quadrants.

For example, calculating \(\cos \frac{5\pi}{4}\) results in \(-\frac{\sqrt{2}}{2}\), demonstrating its third-quadrant position. Understanding cosine behavior is vital for solving problems involving angles and rotations.