Problem 63 - Algebra 2

Question

Find an exact value for sin \(165^{\circ} .\) Show your work.

Step-by-Step Solution

Answer

The exact value of sin \(165^{\circ}\) is \(\frac{\sqrt{6}-\sqrt{2}}{4}\)

Step 1: Express 165 degrees as a sum of 'known' angles

We will express the given angle of 165 degrees as a sum of angles with known trigonometric ratios. Hence, \(165^{\circ}=120^{\circ}+45^{\circ}\)

Step 2: Apply Sum of Angles Identity

We apply the sum of angles identity in trigonometry, which states \(sin(A + B) = sin(A)cos(B) + cos(A)sin(B)\), hence \(sin(165^{\circ})=sin(120^{\circ})cos(45^{\circ})+cos(120^{\circ})sin(45^{\circ})\)

Step 3: Substitute known values

Substitute known values for the sine and cosine functions from the unit circle. Hence, \(sin(165^{\circ})=\frac{\sqrt{3}}{2}*\frac{\sqrt{2}}{2}+-\frac{1}{2}*\frac{\sqrt{2}}{2}=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\)

Step 4: Combine like terms

We can combine like terms to simplify the expression, therefore, \(sin(165^{\circ})=\frac{\sqrt{6}-\sqrt{2}}{4}\)

Key Concepts

Sum of Angles Identity

The sum of angles identity is a crucial concept in trigonometry that helps simplify the sine, cosine, and tangent of angle sums. Specifically, when breaking down angles into two known parts, this identity allows us to calculate the exact values of trigonometric functions for non-standard angles. It makes use of simpler angle values whose sine and cosine are already known, typically these angles are special angles like 30°, 45°, 60°, 90°, and their multiples.
For the sine function, the sum of angles identity is given by the formula:

• \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)

This formula enables us to find the sine of 165° by expressing it as 120° + 45° and using the known sine and cosine values of these angles.
The identity is helpful not only for simplifying calculations but also for understanding the relationships between different trigonometric functions.

Unit Circle

The unit circle is a fundamental resource in trigonometry, representing all possible angles on a circle with a radius of one. It is centered at the origin of a coordinate plane and helps in understanding all trigonometric functions and identities. By using the unit circle, we can easily find the trigonometric ratios for angles, as every point \((x, y)\) on the circle correlates to the cosine and sine of the angle, respectively:

• \( x = \cos(\theta)\)
• \( y = \sin(\theta)\)

Special angles like 30°, 45°, 60°, and 90° have been memorized as they represent simple fractional coordinates on the unit circle.
For example, the unit circle helps quickly identify that \( \cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \) and \( \sin(120^{\circ}) = \frac{\sqrt{3}}{2} \), which we used in solving for \( \sin(165^{\circ}) \). Recognizing these values makes trigonometric problems more approachable and less error-prone.

Trigonometric Ratios

Trigonometric ratios are ratios between the sides of a right triangle. They are fundamental in defining the sine, cosine, and tangent functions for any angle. These functions are directly related to angles and sides of triangles but can also be simplified using the unit circle.

• \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
• \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)

In the unit circle, these ratios align with coordinates. For example, at \(45^{\circ}\), both sine and cosine are \(\frac{\sqrt{2}}{2}\) as shown in the exercise.
Understanding these ratios helps simplify tasks when dealing with angles in both right triangles and the coordinate plane, crucial for finding anyone's trigonometric function values, like sine and cosine.

Sine Function

The sine function is a fundamental trigonometric function that represents the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle. On the unit circle, the sine of an angle corresponds to the y-coordinate of the point where the angle's terminal side intersects the circle. This intuitive representation allows us to understand how the sine varies as the angle changes.

• Sine is positive in the first and second quadrants.
• It has a periodic nature, repeating every \(360^{\circ}\) (or \(2\pi\) radians).
• The maximum value of sine is 1, and the minimum is -1.

In the context of our solution for \(\sin(165^{\circ})\), we recognize that this angle is in the second quadrant, where sine remains positive. Identifying these properties assists in predicting and calculating sine for various angles.