Cofunction Theorem

Definition

The cofunction theorem is a fundamental concept in trigonometry that establishes a relationship between certain trigonometric functions. It states that the trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant are related in a specific way, allowing for the substitution of one function with another in various trigonometric identities and equations.

5 Must Know Facts For Your Next Test

1. The cofunction theorem states that $\sin(x) = \cos(90^\circ - x)$, $\cos(x) = \sin(90^\circ - x)$, and similar relationships for the other trigonometric functions.
2. The cofunction theorem allows for the substitution of one trigonometric function with another, simplifying the calculation of trigonometric expressions.
3. The cofunction theorem is particularly useful in the context of the unit circle, as it describes the relationship between the coordinates of points on the unit circle that are symmetric about the $45^\circ$ line.
4. The cofunction theorem is a key concept in understanding the periodicity and symmetry of trigonometric functions, which is essential for solving a variety of trigonometric problems.
5. The cofunction theorem is often used in conjunction with other trigonometric identities, such as the Pythagorean identities and the reciprocal identities, to simplify and manipulate trigonometric expressions.

Review Questions

• Explain how the cofunction theorem relates the sine and cosine functions.
  • The cofunction theorem states that $\sin(x) = \cos(90^\circ - x)$ and $\cos(x) = \sin(90^\circ - x)$. This means that the sine of an angle is equal to the cosine of the complementary angle (90 degrees minus the original angle), and the cosine of an angle is equal to the sine of the complementary angle. This relationship allows for the substitution of one trigonometric function with another, simplifying the calculation of trigonometric expressions.
• Describe how the cofunction theorem can be used to understand the symmetry of trigonometric functions on the unit circle.
  • The cofunction theorem reveals the symmetry of trigonometric functions on the unit circle. Specifically, the theorem states that points on the unit circle that are symmetric about the $45^\circ$ line have the same sine value but opposite cosine values, and vice versa. This symmetry is a key property of trigonometric functions and is essential for understanding their periodic behavior and solving a variety of trigonometric problems.
• Analyze how the cofunction theorem can be used in conjunction with other trigonometric identities to simplify and manipulate trigonometric expressions.
  • The cofunction theorem is often used in combination with other trigonometric identities, such as the Pythagorean identities and the reciprocal identities, to simplify and manipulate trigonometric expressions. By substituting one trigonometric function for another using the cofunction theorem, and then applying other identities, complex trigonometric expressions can be reduced to simpler forms. This allows for more efficient calculations and a deeper understanding of the relationships between the various trigonometric functions.

Related terms

Trigonometric Functions: The six basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant, which describe the relationships between the sides and angles of a right triangle.

Inverse Trigonometric Functions:

The functions that undo the operations of the basic trigonometric functions, allowing the angle to be found given the value of a trigonometric function.

Trigonometric Identities:

Equations that hold true for all values of the variables involved, allowing for the simplification and manipulation of trigonometric expressions.