Problem 89 - Algebra and Trigonometry

Question

find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$ \sin \frac{11 \pi}{4} \cos \frac{5 \pi}{6}+\cos \frac{11 \pi}{4} \sin \frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

The simplifed expression is \(-\sqrt{6}/4 + \sqrt{2}/4\).

Step 1: Normalize the angles

First step is to normalize the angles, that is, to represent them so they could fit within the range of a standard unit circle angles, which ranges from 0 to \(2\pi\). \(11\pi/4\) is equivalent to \((8\pi + 3\pi) / 4\), or \(2\pi + 3\pi/4\) when simplified. This equates to \(3\pi/4\) on the unit circle. Similarly, \(5\pi/6\) is within the standard range of unit circle.

Step 2: Compute the value of each term independently

Next, we evaluate \(\sin(3\pi / 4)\), \(\cos(3\pi / 4)\), \(\sin(5\pi / 6)\), and \(\cos(5\pi / 6)\) independently. Using the Pythagorean identity, we know that \(\sin(3\pi / 4) = \sqrt{2} / 2\), \(\cos(3\pi / 4) = \sqrt{2} / 2\), \(\sin(5\pi / 6)=1 / 2\) and \(\cos(5\pi / 6)= -\sqrt{3} / 2\).

Step 3: Substitute and Simplify

Now, let's substitute the obtained values into the original expression: \(\sin \frac{11 \pi}{4} \cos \frac{5 \pi}{6}+\cos \frac{11 \pi}{4} \sin \frac{5\pi}{6}\) becomes \(\sqrt{2}/2*(-\sqrt{3}/2) + \sqrt{2}/2*1/2\). Which simplifies to \(-\sqrt{6}/4 + \sqrt{2}/4\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The importance of the unit circle lies in its ability to help us understand the core trigonometric functions: sine, cosine, and tangent.
These functions can be represented geometrically on the unit circle as:

• The x-coordinate of a point on the circle represents the cosine value of the angle.
• The y-coordinate represents the sine value.

The unit circle is usually divided into four quadrants. The angles around the circle range from 0 to \( 2\pi \), covering a complete revolution. Understanding these angles and their corresponding coordinates allows us to calculate any trigonometric function for angle measurements. This becomes exceptionally useful when simplifying or computing expressions manually. Each angle finds its unique representation on the circle, which is why we can simplify calculations, such as finding sine or cosine values, by referring to predefined coordinates and their properties within the unit circle.

Angle Normalization

Angle normalization is an essential process in trigonometry. It involves simplifying an angle to its equivalent value that fits within a specific range, typically from 0 to \( 2\pi \). This is particularly useful when dealing with angles greater than \( 2\pi \) or less than 0.
To normalize an angle like \( \frac{11\pi}{4} \), we subtract full rotations (multiples of \( 2\pi \)) until the resulting angle falls within 0 and \( 2\pi \).
Here's how it works for \( \frac{11\pi}{4} \):

• The angle can be broken down as \( 2\pi + \frac{3\pi}{4} \).
• This simplifies to \( \frac{3\pi}{4} \), an angle that's easier to handle within the unit circle.

By normalizing angles, we avoid unnecessary complications in calculations. This makes trigonometric expressions more manageable, allowing us to easily find ratios for sine and cosine.

Pythagorean Identity

The Pythagorean identity is a crucial concept in trigonometry that relates the squares of sine and cosine functions. The identity is given by:\[\sin^2(\theta) + \cos^2(\theta) = 1\]This equation is derived from the Pythagorean theorem and applies to all angles \( \theta \). It underscores the inseparable bond between the sine and cosine functions as projections of points on the unit circle.
The identity allows us to determine one function value if the other is known. For example, if you know \( \sin(\theta) \), you can find \( \cos(\theta) \) using:\[\cos(\theta) = \sqrt{1 - \sin^2(\theta)}\]Or vice versa. This comes in handy when computing expressions like in our problem. Knowing values for \( \sin(\frac{3\pi}{4}) \) and \( \cos(\frac{3\pi}{4}) \), thanks to the unit circle and Pythagorean identity, simplifies the production of trigonometric expressions.
Remember, these relations only hold for angles within the context of trigonometry on the unit circle and help in understanding the symmetry and properties of trig functions effortlessly.