Problem 26 - Algebra 2 and Trigonometry

Question

In \(24-32,\) find the exact value of each expression. $$ \tan (\arccos 1) $$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Understand the Function

First, comprehend that the expression is \( \tan(\arccos(1)) \). The term \( \arccos(1) \) denotes the angle whose cosine is 1.

Step 2: Determine the Angle

In the unit circle, the angle with a cosine of 1 is 0 radians. Hence, \( \arccos(1) = 0 \).

Step 3: Find the Tangent Value

Now that we know \( \arccos(1) = 0 \), we need to find \( \tan(0) \). The tangent of 0 radians is 0.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps understand how angles relate to different trigonometric functions. Imagine a circle with its center at the origin of a coordinate plane and a radius of 1. This is the unit circle.

• Coordinates on the Unit Circle: Points on this circle correspond to angles measured in radians from the positive x-axis.
• How Angles are Measured: Angles are measured counterclockwise and can be denoted in radians or degrees.
• Relationship with Trigonometric Functions: The x-coordinate of a point on the unit circle represents the cosine of the angle, while the y-coordinate represents the sine.

For example, at an angle of 0 radians, the coordinate is (1,0) because the x-value is 1 and the y-value is 0. Hence, the cosine of 0 radians is 1, reinforcing why on the unit circle, any angle corresponding to the cosine being 1 will have an x-coordinate of 1.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every angle. They allow us to simplify expressions and solve equations more easily.

• Pythagorean Identity: This is one of the most fundamental identities. It states: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] This identity holds true for any angle \(\theta\).
• Angle Sum and Difference Identities: These identities help find the sine, cosine, and tangent of angle sums and differences. They can be crucial for evaluating trigonometric functions not found directly on the unit circle.
• Reciprocal Identities: These relate functions like sine, cosine, and tangent to their reciprocals: cosecant, secant, and cotangent.

Understanding these identities involves recognizing how they simplify complex trigonometric expressions, making it easier to understand properties of angles on the unit circle.

Tangent Function

The tangent function is another primary trigonometric function, often described as the slope of the angle in a right triangle. It's fundamental in various mathematical applications and can be understood very well through the unit circle.

• Definition of Tangent: It is defined as the ratio of sine to cosine for a given angle, specifically: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
• Behavior on the Unit Circle: On the unit circle, tangent is not the x or y coordinate by itself, but rather the slope of the line formed by the radius. This means when moving around the circle, it can take any real value from -∞ to ∞.
• Tangent of Special Angles: At 0 radians, tangent is 0 because the sine is 0 and cosine is 1, resulting in 0. For angles where the cosine is zero, like \(\frac{\pi}{2}\), the tangent function approaches infinity.

The tangent function plays a crucial role in understanding angles and triangles, especially when non-right angles are involved. It provides a deeper connection between the geometry of the circle and linear algebra concepts like slopes and angles.