Sine Sum Identity

Definition

The sine sum identity is a trigonometric identity that relates the sine of the sum or difference of two angles to the sines of the individual angles. It is a fundamental relationship in trigonometry that is widely used in various mathematical applications.

5 Must Know Facts For Your Next Test

1. The sine sum identity states that $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$.
2. The sine difference identity states that $\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)$.
3. The sine sum and difference identities are useful in simplifying trigonometric expressions and solving trigonometric equations.
4. These identities can be derived using the unit circle and the geometric properties of triangles.
5. The sine sum and difference identities are also known as the angle addition and angle subtraction formulas, respectively.

Review Questions

• Explain the relationship between the sine of the sum of two angles and the sines and cosines of the individual angles.
  • The sine sum identity states that $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$. This means that the sine of the sum of two angles is equal to the product of the sine of one angle and the cosine of the other angle, plus the product of the cosine of the first angle and the sine of the second angle. This identity is a fundamental relationship in trigonometry and is widely used in various mathematical applications.
• Describe how the sine difference identity is related to the sine sum identity.
  • The sine difference identity is closely related to the sine sum identity. The sine difference identity states that $\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)$. This formula can be derived from the sine sum identity by substituting $-B$ for $B$. The sine difference identity is also an important trigonometric identity that is used in various mathematical contexts, such as solving trigonometric equations and simplifying trigonometric expressions.
• Analyze the geometric interpretation of the sine sum and difference identities using the unit circle.
  • The sine sum and difference identities can be geometrically interpreted using the unit circle. On the unit circle, the sine of an angle $A$ is represented by the $y$-coordinate of the point on the circle corresponding to that angle. The sine sum identity $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ can be visualized by considering the coordinates of the points on the unit circle representing the angles $A$ and $B$. The sine difference identity $\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)$ can be interpreted in a similar manner, by considering the coordinates of the points representing the angles $A$ and $-B$.

Related terms

Trigonometric Identity: An equation involving trigonometric functions that is true for all values of the variables.

Angle Addition Formula: A formula that relates the sine, cosine, or tangent of the sum or difference of two angles to the sines, cosines, or tangents of the individual angles.

Difference Formula: A formula that relates the trigonometric functions of the difference between two angles to the trigonometric functions of the individual angles.