Problem 188 - Precalculus

Question

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

Step-by-Step Solution

Answer

The exact value of \( \tan \frac{\pi}{4} \) is 1.

Step 1: Understanding the Tangent Function

The tangent (tan) function is a trigonometric function that can be defined as the ratio of the sine of an angle to the cosine of that angle. For an angle \( \theta \), the formula is \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Step 2: Identifying the Special Angle

The angle \( \frac{\pi}{4} \) radians is a special angle known for its easily recognizable sine and cosine values. It is equivalent to 45 degrees.

Step 3: Finding Sine and Cosine Values

For the angle \( \frac{\pi}{4} \), both \( \sin \) and \( \cos \) functions have a value of \( \frac{\sqrt{2}}{2} \). Therefore, \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

Step 4: Applying the Tangent Formula

Use the tangent formula \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Substitute \( \theta = \frac{\pi}{4} \), which gives: \[ \tan\left(\frac{\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \].

Step 5: Final Result

Thus, the exact value of \( \tan\left(\frac{\pi}{4}\right) \) is 1.

Key Concepts

Understanding the Tangent Function

The tangent function is one of the primary trigonometric functions you will frequently encounter in trigonometry. This function is unique because it relates two other trigonometric functions: sine and cosine. Specifically, the tangent of an angle, denoted as \( \tan(\theta) \), is the ratio of the sine of that angle to the cosine of the same angle. To simply put it, the mathematical formula is:\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]This ratio helps us understand the relationship between the angle and the lengths of the sides in a right triangle when dealing with right-angled trigonometry. Remember:

• Tangent is positive in the first and third quadrants of the unit circle.
• Unlike sine and cosine, tangent has asymptotes, where its value becomes undefined.

The tangent function plays a crucial role in solving problems involving angles and their measures, such as the one in our exercise.

The Significance of Special Angles

Special angles are key in trigonometry because they have values that are easy to remember and work with, often involving simple fractions or square roots. The angle \( \frac{\pi}{4} \), or 45 degrees, is one of these special angles. It stands out because the sine and cosine values for this angle are equal, making it quite convenient for calculations involving the tangent function. Working with these angles means:

• Calculations become straightforward.
• You can quickly deduce answers for sine, cosine, and tangent without a calculator.
• They frequently appear in many geometry and trigonometry problems, solidifying their importance in mathematical education.

Understanding special angles can save you time and effort because memorizing their sine and cosine values simplifies solving many standard problems.

Sine and Cosine Values for Special Angles

Sine and cosine functions are foundational in understanding how angles relate to the unit circle. For the special angle \( \frac{\pi}{4} \), the values of \( \sin \) and \( \cos \) are particularly notable since they are equal and both are \( \frac{\sqrt{2}}{2} \). This property simplifies the calculation of the tangent function because:

• At \( \frac{\pi}{4} \), \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \).
• Having memorized these values can accelerate problem-solving speed.

The equal sine and cosine values for \( \frac{\pi}{4} \) encapsulate a symmetrical property of the 45-degree angle, evident in the isosceles right triangle. This angle is often a stepping stone to mastering trigonometric identities and relationships in mathematics.