Problem 21 - Algebra 2 and Trigonometry

Question

In \(18-27\) , express each given function value in terms of a function value of a positive acute angle (the reference angle). \(\cos 312^{\circ}\)

Step-by-Step Solution

Answer

\(\cos 312^{\circ} = \cos 48^{\circ}\).

Step 1: Identify the Quadrant

The angle \(312^{\circ}\) is within the range of \(270^{\circ} \text{ to } 360^{\circ}\), which indicates it lies in the fourth quadrant of the unit circle.

Step 2: Calculate the Reference Angle

The reference angle for an angle in the fourth quadrant is found by subtracting the given angle from \(360^{\circ}\). So, the reference angle \(\theta_r\) is \(360^{\circ} - 312^{\circ} = 48^{\circ}\).

Step 3: Express the Function in Terms of the Reference Angle

In the fourth quadrant, the cosine function is positive. Thus, \(\cos 312^{\circ} = \cos 48^{\circ}\), with \(48^{\circ}\) being the positive acute reference angle.

Key Concepts

Reference Angle

A reference angle is a helpful concept to make understanding angles and trigonometric functions easier. When dealing with any angle on the coordinate plane, rather than using the angle itself to determine trigonometric values, we often use its reference angle. This angle is always measured from the x-axis to the terminal side of the angle, and is always positive and acute.

• A positive acute angle means it is between 0 and 90 degrees.
• No matter which quadrant the original angle is in, the reference angle provides a straightforward way to work with degrees.

To find it, remember: In the fourth quadrant, subtract the angle from 360 degrees. This gives the smallest angle the terminal side has with the x-axis. Using the reference angle can simplify complex problems and is important especially when simplifying trigonometric functions.

Unit Circle

The unit circle is a key concept in trigonometry. This circle has a radius of 1 unit and is centered at the origin of a coordinate plane.It forms the basis for defining trigonometric functions for all angles.Using the unit circle makes it easy to visualize and calculate the values for sine, cosine, and tangent:

• The x-coordinate of a point on the unit circle is the cosine of the angle.
• The y-coordinate is the sine of the angle.
• Being on the unit circle ensures the sum of the squares of these coordinates is always 1 (\( ext{cos}^2(\theta) + ext{sin}^2(\theta) = 1\)).

Choosing angles on the circle and understanding their coordinates helps link angles to their trigonometric values, making it easier to comprehend functions over the different quadrants.

Quadrants

Quadrants divide the coordinate plane into four sections. Knowing which quadrant an angle is in helps determine the signs of its trigonometric functions. Starting from the positive x-axis and moving counterclockwise:

• **First Quadrant**: All trigonometric functions are positive.
• **Second Quadrant**: Sine is positive, while cosine and tangent are negative.
• **Third Quadrant**: Tangent is positive, cosine and sine are negative.
• **Fourth Quadrant**: Cosine is positive, sine and tangent are negative.

Understanding which quadrant the terminal side of an angle lies in is crucial for correctly solving trigonometric problems, as it determines whether the function values are positive or negative. This understanding is key to computing values effectively and avoiding common mistakes.