Complementary Angles

Definition

Complementary angles are two angles that add up to 90 degrees. They are angles that, when combined, form a right angle or a square corner. This concept is essential in understanding various trigonometric identities and formulas related to angles.

5 Must Know Facts For Your Next Test

1. Complementary angles are often represented using the symbol $\perp$, which indicates that the angles are perpendicular to each other.
2. The sum of complementary angles is always 90 degrees, or $\pi/2$ radians.
3. Complementary angles are crucial in understanding the trigonometric identities related to the sum and difference of angles, as well as the double-angle, half-angle, and reduction formulas.
4. Identifying complementary angles is important in solving various geometric problems, such as finding missing angles in triangles or quadrilaterals.
5. Complementary angles can be used to simplify trigonometric expressions and equations, as the trigonometric functions of complementary angles are related in specific ways.

Review Questions

• Explain how the concept of complementary angles is applied in the context of the Sum and Difference Identities (Topic 7.2).
  • The concept of complementary angles is crucial in understanding the Sum and Difference Identities, which relate the trigonometric functions of the sum or difference of two angles. These identities rely on the fact that the sum of two complementary angles is 90 degrees. For example, the identity $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ can be derived by considering the complementary angles $A$ and $90 - B$.
• Describe how the properties of complementary angles are used in the derivation of the Double-Angle, Half-Angle, and Reduction Formulas (Topic 7.3).
  • The properties of complementary angles are fundamental in the derivation of the Double-Angle, Half-Angle, and Reduction Formulas. For instance, the Double-Angle Formula $\sin(2A) = 2\sin(A)\cos(A)$ can be derived by considering the complementary angles $A$ and $90 - A$, and applying the Sum and Difference Identities. Similarly, the Half-Angle Formula $\sin(A/2) = \pm\sqrt{(1 - \cos(A))/2}$ relies on the fact that the half-angle $A/2$ is complementary to $(90 - A)/2$.
• Analyze how the concept of complementary angles is applied in the context of the Angles topic (Topic 5.1) to solve various geometric problems.
  • In the Angles topic, the understanding of complementary angles is crucial for solving geometric problems involving angles. For example, when determining the missing angles in a triangle or a quadrilateral, the fact that the sum of the angles in a triangle is 180 degrees, and the sum of the angles in a quadrilateral is 360 degrees, relies on the properties of complementary angles. By identifying complementary angles, one can set up equations to solve for the unknown angles, which is essential in various geometric proofs and constructions.

Related terms

Supplementary Angles: Supplementary angles are two angles that add up to 180 degrees. They are angles that, when combined, form a straight line.

Acute Angle: An acute angle is an angle that measures less than 90 degrees. Complementary angles always include at least one acute angle.

Obtuse Angle: An obtuse angle is an angle that measures more than 90 degrees. Complementary angles can never include an obtuse angle.