Problem 337 - Precalculus

Question

For the following exercises, find the exact value of the given expression. $$ \tan \frac{\pi}{4} $$

Step-by-Step Solution

Answer

The exact value is 1.

Step 1: Identify the Problem

We need to find the exact value of the trigonometric expression \( \tan \frac{\pi}{4} \).

Step 2: Recall Important Trigonometric Values

Remember that in trigonometry, specific angles have known values for sine, cosine, and tangent. The angle \( \frac{\pi}{4} \) radians is equivalent to 45 degrees, a commonly used angle.

Step 3: Understand the Relationship

For the angle \( \frac{\pi}{4} \), the sine and cosine values are equal. This is because both sides of a 45-degree right triangle (which corresponds to \( \frac{\pi}{4} \) radians) are equal, so \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 4: Apply the Definition of Tangent

The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Step 5: Calculate \( \tan \frac{\pi}{4} \)

Substitute the values of sine and cosine for \( \frac{\pi}{4} \): \( \tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \).

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions in mathematics. It is often used to find the ratio of two sides of a right triangle. The tangent of an angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. This can also be expressed using the sine and cosine functions:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

When you hear about executing trig operations, the tangent is crucial in many practical applications like physics problems or surveying. In this context, calculating the tangent involves understanding its relationship with the other trigonometric functions, especially sine and cosine. For our specific exercise, knowing that \( \tan \frac{\pi}{4} = 1 \) comes from recognizing that at this particular angle, the sine and cosine are equal.

Unit Circle

The unit circle is a very helpful concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle allows us to represent trigonometric functions geometrically. Here's how it works:

• The angle \( \theta \) in the unit circle is measured from the positive x-axis.
• The coordinates of a point on the circle, represented as \((x, y)\), are equivalent to \( (\cos \theta, \sin \theta) \).

Using the unit circle, you can easily determine the sine and cosine values for any key angles, like \( \frac{\pi}{4} \), where both coordinates are \( \frac{\sqrt{2}}{2} \). This visual tool simplifies understanding how these trigonometric functions behave and interact. By using the unit circle, you also see why \( \tan \frac{\pi}{4} \) equals 1, as both sine and cosine are the same, making their ratio equal to one.

Exact Values

Exact values in trigonometry refer to the precise, often non-decimalized values of trigonometric functions at standard angles. Learning these by heart greatly simplifies solving numerous math problems. They are the foundational values for sine, cosine, and tangent at common angles like 0, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \).
The expression \( \tan \frac{\pi}{4} = 1 \) is an example of an exact value. This is important because it reinforces the understanding of key trigonometric relationships:

• Knowing exact values means less reliance on calculators.
• It helps in algebraic transformations or solving complex equations quickly.

By memorizing the exact values of trigonometric functions at these key angles, you can confidently tackle a wide range of mathematical challenges without getting bogged down by decimal approximations.