Problem 27 - Algebra 2 and Trigonometry

Question

In \(26-33 :\) a. Rewrite each function value in terms of its cofunction. b. Find the exact value of the function value found in a. $$ \cos \frac{\pi}{4} $$

Step-by-Step Solution

Answer

\( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)

Step 1: Recall the Cofunction Identity

The cofunction identity for cosine is: \[ \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \] This tells us that the cosine of an angle is equal to the sine of its complementary angle.

Step 2: Apply the Cofunction Identity

Given the function \( \cos \frac{\pi}{4} \), apply the identity: \[ \cos \frac{\pi}{4} = \sin\left(\frac{\pi}{2} - \frac{\pi}{4}\right) \] Simplify the expression inside the sine function.

Step 3: Simplify the Angle

Simplify the expression \( \frac{\pi}{2} - \frac{\pi}{4} \):\[ \frac{\pi}{2} - \frac{\pi}{4} = \frac{2\pi}{4} - \frac{\pi}{4} = \frac{\pi}{4} \]Thus, we have: \[ \sin\left(\frac{\pi}{4}\right) \]

Step 4: Find the Exact Value

Now, calculate the exact value using known trigonometric values: \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \] This is a common angle value for sine.

Key Concepts

Trigonometric Functions

Trigonometric functions are the building blocks of trigonometry. They are essential for understanding the relationships between angles and sides in right-angled triangles. There are six primary trigonometric functions:

• Sine (\( \sin \theta \))
• Cosine (\( \cos \theta \))
• Tangent (\( \tan \theta \))
• Cosecant (\( \csc \theta \))
• Secant (\( \sec \theta \))
• Cotangent (\( \cot \theta \))

These functions allow us to solve triangles, find angles, and determine exact values for specific angles. Each function relates to a specific ratio of a right triangle's sides. For example, cosine measures the ratio of the adjacent side to the hypotenuse.
These functions are also periodic and can model real-world phenomena like sound waves and light patterns.

Complementary Angles

Complementary angles are two angles whose measures add up to \( \frac{\pi}{2} \) radians or 90 degrees. This is a central concept in trigonometry because of the relationship these angles have in forming right angles.
When two angles are complementary, the sine of one angle equals the cosine of the other angle and vice versa. This relationship, known as the cofunction identity, is crucial for solving various trigonometric problems.

• If \( \theta \) is an angle, then its complementary angle can be found using \( \frac{\pi}{2} - \theta \).
• The cofunction identity: \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta) \).

This identity helps simplify expressions by changing a cosine problem into a sine problem or vice versa.

Exact Values in Trigonometry

Finding exact values in trigonometry requires knowledge of special angles and their sine, cosine, and tangent values. Special angles like \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \) have specific trigonometric values that are often memorized.

• \( \sin(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• \( \sin(\frac{\pi}{6}) = \frac{1}{2} \)
• \( \cos(\frac{\pi}{3}) = \frac{1}{2} \)

These values are derived from the unit circle, where each point represents an angle corresponding to its sine and cosine values.
Understanding these special angles can greatly simplify trigonometric calculations and make evaluating expressions more intuitive.

Cosine Function

The cosine function, denoted as \( \cos(\theta) \), is one of the fundamental trigonometric functions. It measures the ratio of the adjacent side to the hypotenuse in a right triangle. This function is periodic, which means it repeats its values in regular intervals—specifically, every \(2\pi\) radians or 360 degrees.The cosine function is instrumental in modeling oscillatory behavior such as sound waves or other harmonic motions. Its graph is a wave that starts from 1, dips to -1, and returns to 1 as it completes a cycle.

• The function is even, meaning \( \cos(-\theta) = \cos(\theta) \).
• Key values include \( \cos(0) = 1 \) and \( \cos(\frac{\pi}{2}) = 0 \).

With its diverse applications, the cosine function is foundational for many areas of mathematics and engineering.