Problem 328 - Precalculus

Question

For the following exercises, draw the angle provided in standard position on the Cartesian plane. Find the exact value of \(\cos \frac{\pi}{4}\)

Step-by-Step Solution

Answer

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

Step 1: Sketch the Angle

To sketch the angle \( \frac{\pi}{4} \), recall that angles in standard position start from the positive x-axis. The angle \( \frac{\pi}{4} \) radians corresponds to 45 degrees. Draw a line from the origin at a 45-degree angle from the positive x-axis.

Step 2: Identify the Reference Triangle

The angle \( \frac{\pi}{4} \) is in the first quadrant where both x and y coordinates are positive. Draw a line perpendicular to the x-axis to form a right triangle. This triangle is an isosceles right triangle, meaning its two non-hypotenuse sides are equal, each representing the legs of length 1.

Step 3: Determine the Hypotenuse

Use the Pythagorean theorem to find the hypotenuse of the 45-degree triangle. If both legs are 1, then the hypotenuse is \( \sqrt{2} \), calculated as \( \sqrt{1^2 + 1^2} = \sqrt{2} \).

Step 4: Calculate \( \cos \frac{\pi}{4} \)

The cosine of an angle in the unit triangle is the ratio of the adjacent side to the hypotenuse. For the angle \( \frac{\pi}{4} \), both the adjacent side and the opposite side are 1, and the hypotenuse is \( \sqrt{2} \). Therefore, \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \).

Step 5: Simplify the Expression

Simplify \( \frac{1}{\sqrt{2}} \) by rationalizing the denominator. Multiply both the numerator and the denominator by \( \sqrt{2} \), resulting in \( \frac{\sqrt{2}}{2} \). Thus, \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in geometry, and are often represented using a unit circle. These functions include sine, cosine, and tangent, and they help us understand the relation between angles and side lengths in right triangles. One of the primary uses of trigonometric functions is to relate an angle of a right triangle to the ratio of the sides of the triangle. In a unit circle, where the radius is 1, these ratios can be easily visualized and calculated.

• Sine (sin): The ratio of the opposite side to the hypotenuse.
• Cosine (cos): The ratio of the adjacent side to the hypotenuse.
• Tangent (tan): The ratio of the opposite side to the adjacent side.

For example, if you need to find the cosine of a 45-degree angle using trigonometric functions, you can visualize it as an angle in a right triangle formed within a unit circle. The cosine is then calculated by dividing the length of the adjacent side by the length of the hypotenuse.

Since trigonometric functions are periodic, they repeat their values in a predictable pattern, which is why they are so useful in understanding phenomena that have a cyclical nature.

Reference Angle

A reference angle is crucial when dealing with trigonometric functions, as it simplifies the process of finding the trigonometric values of angles, especially when you are working outside the first quadrant. It is the smallest angle formed from the terminal side of an angle in standard position to the x-axis.

To find the reference angle:

• If the angle is in the first quadrant, the reference angle is the angle itself.
• If the angle is in the second or third quadrant, subtract it from \( \pi \) or 180 degrees.
• If the angle is in the fourth quadrant, subtract 360 degrees or \( 2\pi \) from it.

The reference angle makes complex trigonometric problems often look simpler by bringing them back to the much easier-to-handle first quadrant.

Right Triangle

A right triangle is a triangle in which one of the angles is exactly 90 degrees. This makes it very special in terms of geometric properties and calculations. Right triangles play a central role in trigonometry. Their recognizable property is the Pythagorean theorem, which states:\[ a^2 + b^2 = c^2 \]where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse.

In the context of trigonometry:

• The sides opposite, adjacent, and hypotenuse are used to calculate sine, cosine, and tangent.
• A right triangle can be formed in the coordinate plane by drawing a perpendicular line from a point to the x-axis, helping to visualize angles in standard position.

The most common special right triangles used in trigonometric calculations are the 45-45-90 and 30-60-90 triangles. These have known side ratios which make trigonometric calculations straightforward. For example, in a 45-45-90 triangle, if the lengths of the two legs are 1, then the hypotenuse is \sqrt{2}\, perfectly reflecting the ratio used in calculating the cosine of a 45-degree angle.