Problem 94 - Algebra and Trigonometry

Question

let $$ f(x)=\sin x, g(x)=\cos x, \text { and } h(x)=2 x $$ Find the exact value of each expression. Do not use a calculator. $$ g\left(\frac{5 \pi}{6}+\frac{\pi}{6}\right)+g\left(\frac{5 \pi}{6}\right)+g\left(\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

-1

Step 1: Simplify the Radian Expression

Start by simplifying the radian expressions. Add up \( \frac{5 \pi}{6} \) and \( \frac{\pi}{6} \) to get \( \pi \).

Step 2: Evaluate the Cosine Function

Evaluate the function \( g(x) = \cos(x) \) at \( x = \pi, \frac{5 \pi}{6}, \) and \( \frac{\pi}{6} \). Using the unit circle, we know that \( \cos(\pi) = -1 \), \( \cos\left(\frac{5 \pi}{6}\right) = -\frac{\sqrt{3}}{2} \), and \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)

Step 3: Add up the Results

Finally, add up the values obtained from evaluating the function at the given inputs. The sum is \( -1 - \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \). The \( - \frac{\sqrt{3}}{2} \) and \( \frac{\sqrt{3}}{2} \) cancel out each other to give the final result of -1.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 unit centered at the origin of the coordinate plane.
The circumference of the unit circle represents angles measured in radians. The unit circle helps visualize sine, cosine, and tangent values as coordinates on the circle.Key Points of the Unit Circle:

• Every point on the unit circle can be expressed as \(( ext{cos} heta, ext{sin} heta)\), where \( heta \) is the angle in radians.
• The cosine of \( heta \) is the x-coordinate, and the sine of \( heta \) is the y-coordinate.
• Due to its geometry, the unit circle is symmetric, making it a handy tool for finding trigonometric values at various angles.
• Angles in different quadrants of the circle result in sine and cosine values being positive or negative. For example, in the second quadrant, cosine is negative, but sine is positive.

In trigonometric calculations, such as those in the original exercise, values like \( rac{5 ext{π}}{6} \), \( rac{ ext{π}}{6} \), and \( ext{π} \) can be quickly evaluated using the unit circle. Understanding this concept is crucial for solving trigonometric problems efficiently.

Cosine Function

The cosine function, one of the primary trigonometric functions, describes the relationship between an angle and the lengths of the sides of a right triangle. In the context of the unit circle, it provides the x-coordinate of a point on the circle.Cosine Function Characteristics:

• It is an even function, meaning \( ext{cos}(- heta) = ext{cos}( heta) \).
• The function is periodic with a period of \( 2 ext{π} \), so \( ext{cos}( heta + 2 ext{π}) = ext{cos}( heta) \).
• Values range from -1 to 1, inclusive.

To evaluate the cosine of specific angles, like those given in the exercise, you can reference the known cosine values from the unit circle. For example:

• \( ext{cos}( ext{π}) = -1 \)
• \( ext{cos}ig(rac{5 ext{π}}{6}ig) = -rac{ ext{√3}}{2} \)
• \( ext{cos}ig(rac{ ext{π}}{6}ig) = rac{ ext{√3}}{2} \)

Learning and remembering these values helps in quickly solving trigonometric problems.

Radian Measure

Radian measure is an alternative unit of measure for angles. Unlike degrees, which divide a circle into 360 parts, radians divide the circle based on the radius length.
Specifically, one radian is the angle created when the arc length is equal to the radius of the circle.Why Radians?

• Simpler and more natural for mathematical calculations, especially in calculus.
• Directly relates angle measures to measurements of circles, making it ideal for trigonometric applications.
• One full circle (360 degrees) is \( 2 ext{π} \) radians. Thus, \( ext{π} \) radians is equal to 180 degrees.

By using radian measure, angles such as \( rac{5 ext{π}}{6} \) and \( rac{ ext{π}}{6} \) provide a more straightforward approach to calculate trigonometric functions like cosine without a calculator. Understanding radian measure facilitates a deeper comprehension of angle-related computations and simplifies the process of solving trigonometric expressions.