Tangent

Definition

The tangent is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side. This function is pivotal in connecting angle measures to side lengths in triangles, and it also plays a key role in the unit circle, where the tangent function represents the slope of the line that intersects the unit circle at a given angle.

5 Must Know Facts For Your Next Test

1. The tangent function is defined as $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$, making it essential for solving right triangles.
2. The tangent function has a period of $$\pi$$ radians, meaning it repeats its values every $$\pi$$ radians.
3. Tangent is undefined for angles where the cosine equals zero, specifically at odd multiples of $$\frac{\pi}{2}$$, where it has vertical asymptotes.
4. On the unit circle, the value of tangent at an angle is equal to the y-coordinate divided by the x-coordinate, or $$\tan(\theta) = \frac{y}{x}$$.
5. Tangent values can be determined using sine and cosine as $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$.

Review Questions

• How does the tangent function relate to other trigonometric functions like sine and cosine?
  • The tangent function is deeply connected to sine and cosine through its definition as $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$. This means that knowing the values of sine and cosine for a given angle allows you to easily find the tangent. Additionally, understanding these relationships helps in solving problems involving right triangles, as each function provides different ratios relevant to different sides.
• Why is it important to recognize where tangent is undefined and how does this relate to its graph?
  • Tangent is undefined at angles where cosine equals zero, specifically at odd multiples of $$\frac{\pi}{2}$$. This causes vertical asymptotes in its graph, which indicates that as you approach these angles, the value of tangent increases or decreases without bound. Recognizing these points is crucial for sketching accurate graphs and understanding tangent's behavior around those angles.
• Evaluate how understanding the properties of tangent can aid in solving real-world problems involving angles and distances.
  • Understanding tangent's properties allows you to model real-world situations such as determining heights or distances using angles measured from a point. For instance, if you know an angle from a point on level ground to the top of a tree, you can use tangent to find the height by applying $$h = d \cdot \tan(\theta)$$ where $$h$$ is height and $$d$$ is distance from the base. This relationship illustrates how trigonometric functions like tangent bridge theoretical mathematics with practical applications.

Related terms

sine: A trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.

cosine: A trigonometric function that represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

unit circle: A circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions for all angles.