Problem 13 - Precalculus with Limits

Question

In Exercises 13 - 28, find the exact values of the sine, cosine, and tangent of the angle. \( \dfrac{11\pi}{12} = \dfrac{3\pi}{4} + \dfrac{\pi}{6} \)

Step-by-Step Solution

Answer

The exact trigonometric values for the angle \( \dfrac{11\pi}{12}\) are: \( \sin(\dfrac{11\pi}{12}) = \dfrac{\sqrt{6}+\sqrt{2}}{4}\), \( \cos(\dfrac{11\pi}{12}) = \dfrac{-\sqrt{6}+\sqrt{2}}{4}\), and \( \tan(\dfrac{11\pi}{12})= -1\).

Step 1: Apply Addition Formula for Sine

We are using the following formula for sine of a sum of angles: \(\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)\). Thus we get, \(\sin(\dfrac{11\pi}{12}) = \sin(\dfrac{3\pi}{4} + \dfrac{\pi}{6}) = \sin(\dfrac{3\pi}{4})\cos(\dfrac{\pi}{6}) + \cos(\dfrac{3\pi}{4})\sin(\dfrac{\pi}{6})\).

Step 2: Calculate the Sine Function

First, find the sine values for \( \dfrac{3\pi}{4}\) and \( \dfrac{\pi}{6}\). Here, \(\sin(\dfrac{3\pi}{4})=\dfrac{\sqrt{2}}{2}\) and \(\sin(\dfrac{\pi}{6})=\dfrac{1}{2}\). After that, calculate the cosine values. Note that \(\cos(\dfrac{3\pi}{4})=-\dfrac{\sqrt{2}}{2}\) and \(\cos(\dfrac{\pi}{6})=\dfrac{\sqrt{3}}{2}\). Substituting these values in, we get \(\sin(\dfrac{11\pi}{12})=\dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2}=\dfrac{\sqrt{6}+\sqrt{2}}{4}\).

Step 3: Apply Addition Formula for Cosine

We have a similar formula for cosine of a sum of angles: \(\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)\). So, we get \(\cos(\dfrac{11\pi}{12})=\cos(\dfrac{3\pi}{4} + \dfrac{\pi}{6}) = \cos(\dfrac{3\pi}{4})\cos(\dfrac{\pi}{6}) - \sin(\dfrac{3\pi}{4})\sin(\dfrac{\pi}{6})\).

Step 4: Calculate the Cosine Function

Use the previous sine and cosine values and plug them in to get \(\cos(\dfrac{11\pi}{12})=-\dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2}-\dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2}=\dfrac{-\sqrt{6}+\sqrt{2}}{4}\).

Step 5: Calculate Tangent using Sine and Cosine

The tangent of an angle is the ratio of the sine to the cosine of that angle. So, we compute \(\tan(\dfrac{11\pi}{12}) = \dfrac{\sin(\dfrac{11\pi}{12})}{\cos(\dfrac{11\pi}{12})} = \dfrac{\dfrac{\sqrt{6}+\sqrt{2}}{4}}{\dfrac{-\sqrt{6}+\sqrt{2}}{4}} = -1\).

Key Concepts

Sine Function

The sine function, denoted as \( \sin \) is a fundamental trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For an angle \( \theta \) within a unit circle, the \( \sin(\theta) \) value can be visually interpreted as the y-coordinate of the point where the terminal side of \( \theta \) intersects the circle.

When learning about the sine function, it's vital to be familiar with its values at specific standard angles, such as \( \frac{\pi}{6} \) (30 degrees) and \( \frac{\pi}{4} \) (45 degrees) or \( \frac{3\pi}{4} \) (135 degrees), because these values are often used in solving problems involving trigonometric addition formulas.

Cosine Function

The cosine function is another key trigonometric function, notated as \( \cos \). In a right triangle, it represents the ratio of the adjacent side over the hypotenuse. When dealing with the unit circle, \( \cos(\theta) \) corresponds to the x-coordinate of the point at which the terminal side of the angle \( \theta \) intersects the circle.

Important Values

Knowing the values of \( \cos \) at certain angles is crucial. For instance, \( \cos(\frac{\pi}{6}) \) equals \( \frac{\sqrt{3}}{2} \) and the \( \cos(\frac{3\pi}{4}) \) equals \( -\frac{\sqrt{2}}{2} \). These are essential for calculations in trigonometric equations and particularly for working with addition formulas.

Tangent Function

Tangent is the third primary trigonometric function, indicated by \( \tan \). Unlike sine and cosine, tangent is the ratio of the opposite side over the adjacent side in a right-angled triangle. In terms of the unit circle, \( \tan(\theta) \) is not directly represented as a coordinate, but rather as the slope of the terminal side of the angle \( \theta \) or the y-value divided by the x-value at the intersection on the unit circle.

Ratio of Sine and Cosine

The tangent can also be understood as the ratio of the sine function to the cosine function, making it critical to understand both \( \sin \) and \( \cos \) when working with \( \tan \) in trigonometric identities and equations.

Angle Sum Identity

Angle sum identities are a cornerstone of trigonometry. These formulas express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For instance, the angle sum identity for sine is \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \), and for cosine, it is \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \).

Practical Application

These identities are immensely valuable in solving trigonometric expressions that involve the angles being added or subtracted, which is evident in the steps given to find the exact trigonometric values of \( \frac{11\pi}{12} \) by breaking it into \( \frac{3\pi}{4} \) and \( \frac{\pi}{6} \) and then applying the angle sum identities for sine and cosine. Understanding these identities allows for the simplification of complex trigonometric expressions and the solving of equations.