Trigonometric identities

Definition

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables within their domains. They are essential tools for simplifying expressions and solving trigonometric equations.

5 Must Know Facts For Your Next Test

1. The Law of Cosines can be derived using the Pythagorean Theorem and trigonometric identities.
2. $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$ is a commonly used double-angle identity.
3. Sum and difference formulas, such as $\cos(a \pm b)$ and $\sin(a \pm b)$, simplify complex trigonometric expressions.
4. Trigonometric identities like $1 + \cot^2(x) = \csc^2(x)$ are used to transform non-right triangle problems into solvable forms.
5. Verifying trigonometric identities often involves algebraic manipulation, including factoring and combining fractions.

Review Questions

• How can you derive the Law of Cosines using trigonometric identities?
• What is the double-angle formula for cosine?
• Why are sum and difference formulas important in simplifying trigonometric expressions?

Related terms

Law of Cosines:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$, relates the lengths of sides in any triangle to the cosine of one of its angles.

Pythagorean Identity:

$\sin^2(x) + \cos^2(x) = 1$, fundamental identity linking sine and cosine.

Double-Angle Formula: $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$, expresses trigonometric functions at double angles.