Trigonometric functions

Definition

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in studying periodic phenomena and in calculus for analyzing oscillatory behaviors.

5 Must Know Facts For Your Next Test

1. The primary trigonometric functions are sine ($\sin$), cosine ($\cos$), and tangent ($\tan$).
2. $\sin(\theta) = \frac{opposite}{hypotenuse}$, $\cos(\theta) = \frac{adjacent}{hypotenuse}$, and $\tan(\theta) = \frac{opposite}{adjacent}$. These ratios define the basic trigonometric functions.
3. The unit circle is a key concept for understanding trigonometric functions, where $x$-coordinates represent $\cos(\theta)$ and $y$-coordinates represent $\sin(\theta)$.
4. Trigonometric identities such as $\sin^2(x) + \cos^2(x) = 1$ are crucial for simplifying expressions and solving equations.
5. Inverse trigonometric functions (like $\arcsin$, $\arccos$, and $\arctan$) are used to find angles when given function values.

Review Questions

• What are the definitions of sine, cosine, and tangent in terms of a right triangle?
• How does the unit circle help in understanding the values of trigonometric functions?
• State and prove one fundamental trigonometric identity.

Related terms

Unit Circle:

A circle with a radius of one centered at the origin on the coordinate plane, used to define sine, cosine, and tangent.

Pythagorean Identity: $\sin^2(x) + \cos^2(x) = 1$, an important equation relating $\sin$ and $\cos$.

Inverse Trigonometric Functions:

Functions like $\arcsin$, $\arccos$, and $\arctan$ that determine the angle corresponding to a given trigonometric value.