Cofunction Identities

Definition

Cofunction identities are mathematical relationships that exist between certain trigonometric functions, such as sine, cosine, tangent, and their reciprocal functions (cosecant, secant, and cotangent). These identities allow for the conversion between different trigonometric functions, providing a useful tool for simplifying and solving trigonometric expressions.

5 Must Know Facts For Your Next Test

1. Cofunction identities are particularly useful when working with the sum and difference identities, as they allow for the conversion between different trigonometric functions within the same expression.
2. The cofunction identities for the sine and cosine functions are $\sin(90^\circ - x) = \cos(x)$ and $\cos(90^\circ - x) = \sin(x)$.
3. The cofunction identities for the tangent and cotangent functions are $\tan(90^\circ - x) = \cot(x)$ and $\cot(90^\circ - x) = \tan(x)$.
4. Cofunction identities can be used to simplify trigonometric expressions, especially when dealing with complementary angles (angles that add up to $90^\circ$).
5. Understanding cofunction identities is crucial for solving a variety of trigonometric equations and inequalities, as well as for evaluating trigonometric functions.

Review Questions

• Explain how cofunction identities can be used to simplify trigonometric expressions involving complementary angles.
  • Cofunction identities allow for the conversion between trigonometric functions involving complementary angles (angles that add up to $90^\circ$). For example, using the identity $\sin(90^\circ - x) = \cos(x)$, a trigonometric expression like $\sin(60^\circ) + \cos(30^\circ)$ can be simplified to $\cos(30^\circ) + \cos(30^\circ) = 2\cos(30^\circ)$. This simplification is possible because $60^\circ$ and $30^\circ$ are complementary angles, and the cofunction identity allows for the substitution of one trigonometric function for another.
• Describe how cofunction identities can be used to solve trigonometric equations and inequalities.
  • Cofunction identities can be used to transform trigonometric equations and inequalities into a more manageable form. For instance, when solving an equation like $\tan(x) = \cot(y)$, the cofunction identity $\tan(90^\circ - x) = \cot(x)$ can be used to rewrite the equation as $\tan(90^\circ - x) = \tan(y)$, which can then be solved for $x$ or $y$ using other trigonometric identities and techniques. Similarly, cofunction identities can be used to simplify trigonometric inequalities, making it easier to determine the solution set.
• Analyze the relationship between cofunction identities and inverse trigonometric functions, and explain how this relationship can be leveraged in problem-solving.
  • Cofunction identities are closely related to inverse trigonometric functions, as they both involve the conversion between different trigonometric functions. For example, the identity $\sin(90^\circ - x) = \cos(x)$ can be rewritten as $\arcsin(x) = \frac{\pi}{2} - \arccos(x)$, highlighting the connection between cofunction identities and inverse trigonometric functions. This relationship can be leveraged in problem-solving by using cofunction identities to transform expressions involving inverse trigonometric functions into a more manageable form, or by using inverse trigonometric functions to find the angle given the value of a trigonometric function and the cofunction identity.

Related terms

Reciprocal Identities: Reciprocal identities express the relationship between a trigonometric function and its reciprocal function, such as $\sin(x) = \frac{1}{\csc(x)}$.

Pythagorean Identities: Pythagorean identities are relationships between the trigonometric functions that are based on the properties of right triangles, such as $\sin^2(x) + \cos^2(x) = 1$.

Inverse Trigonometric Functions: Inverse trigonometric functions, such as $\arcsin(x)$ and $\arccos(x)$, are used to find the angle given the value of a trigonometric function, and are closely related to cofunction identities.