key term - Supplementary Angles

Definition

Supplementary angles are two angles that, when added together, equal 180 degrees. They are angles that form a straight line or a flat surface. This concept is important in understanding the relationships between angles and their properties, which is crucial in the context of topics such as 5.1 Angles, 7.1 Solving Trigonometric Equations with Identities, and 7.3 Double-Angle, Half-Angle, and Reduction Formulas.

5 Must Know Facts For Your Next Test

1. Supplementary angles are useful in solving trigonometric equations by allowing you to find the reference angle or coterminal angles.
2. The sum of the double-angle and half-angle formulas for sine and cosine are based on the properties of supplementary angles.
3. Supplementary angles are important in understanding the behavior of trigonometric functions, as they can be used to simplify or rewrite expressions.
4. Knowing the properties of supplementary angles can help in the reduction of trigonometric expressions, which is a key skill in solving trigonometric equations.
5. Supplementary angles play a role in the trigonometric identities used to solve trigonometric equations, as they allow for the manipulation of the angles involved.

Review Questions

• Explain how supplementary angles can be used to solve trigonometric equations.
  • Supplementary angles can be used to solve trigonometric equations by allowing you to find the reference angle or coterminal angles. For example, if you have an equation like $\sin(\theta) = 0.5$, you can use the fact that $\sin(\theta) = \sin(180^\circ - \theta)$ to find two possible solutions for $\theta$. This property of supplementary angles is crucial in solving trigonometric equations, as it provides a way to simplify and manipulate the angles involved.
• Describe the relationship between supplementary angles and the double-angle and half-angle formulas for sine and cosine.
  • The sum of the double-angle and half-angle formulas for sine and cosine are based on the properties of supplementary angles. For example, the double-angle formula for sine states that $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. This formula is derived from the fact that $\sin(2\theta) = \sin(180^\circ - 2\theta)$, which are supplementary angles. Understanding the connection between supplementary angles and these important trigonometric formulas is crucial for manipulating and simplifying trigonometric expressions.
• Analyze how the properties of supplementary angles can be used to reduce trigonometric expressions and solve trigonometric equations.
  • The properties of supplementary angles can be used to significantly simplify and reduce trigonometric expressions, which is a key skill in solving trigonometric equations. By recognizing that $\sin(\theta) = \sin(180^\circ - \theta)$ and $\cos(\theta) = -\cos(180^\circ - \theta)$, you can manipulate the angles involved in an expression to make it easier to evaluate or solve. This understanding of supplementary angles allows you to rewrite trigonometric expressions in a more manageable form, which is essential in the context of solving trigonometric equations using identities and other techniques.

Related terms

Complementary Angles:

Complementary angles are two angles that, when added together, equal 90 degrees. They are angles that form a right angle.

Vertical Angles: Vertical angles are two non-adjacent angles formed by the intersection of two lines. They are always supplementary angles.

Linear Pair: A linear pair is two adjacent angles that form a straight line, and therefore are supplementary angles.