Double-angle formulas are trigonometric identities that express the sine, cosine, and tangent of twice an angle in terms of the sine and cosine of the original angle. These formulas are crucial for verifying trigonometric identities and simplifying trigonometric expressions involving double-angle terms.
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The double-angle formulas for sine, cosine, and tangent are: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$, and $\tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}$.
Double-angle formulas are particularly useful for verifying trigonometric identities, as they allow you to rewrite expressions involving double-angle terms in a simpler form.
When simplifying trigonometric expressions, double-angle formulas can be used to combine or separate terms involving double-angle functions.
Double-angle formulas are also used in conjunction with angle addition formulas to derive other trigonometric identities, such as the half-angle formulas.
Understanding and applying double-angle formulas is essential for solving a wide range of trigonometric problems, from evaluating trigonometric functions to solving trigonometric equations.
Review Questions
Explain how the double-angle formulas for sine, cosine, and tangent are derived.
The double-angle formulas for sine, cosine, and tangent are derived using the fundamental trigonometric identities and the definitions of the trigonometric functions. For example, the formula for $\sin(2\theta)$ can be derived by applying the angle addition formula for sine: $\sin(\theta + \theta) = \sin(\theta)\cos(\theta) + \cos(\theta)\sin(\theta)$, which simplifies to $2\sin(\theta)\cos(\theta)$. Similarly, the formulas for $\cos(2\theta)$ and $\tan(2\theta)$ can be derived using the angle addition formulas and the Pythagorean identity $\cos^2(\theta) + \sin^2(\theta) = 1$.
Describe how double-angle formulas can be used to verify trigonometric identities.
To verify a trigonometric identity using double-angle formulas, you would first need to rewrite any terms involving double-angle functions using the appropriate double-angle formula. This allows you to express the identity in terms of the sine and cosine of the original angle, which can then be simplified and compared to the other side of the identity. By using the double-angle formulas to manipulate the expression, you can demonstrate that the two sides of the identity are equivalent, thereby verifying the identity.
Analyze how double-angle formulas can be applied to simplify trigonometric expressions.
Double-angle formulas can be used to simplify trigonometric expressions in a few key ways. First, they allow you to combine or separate terms involving double-angle functions, making the expression easier to work with. Second, they can be used in conjunction with other trigonometric identities, such as the Pythagorean identities, to further simplify the expression. Finally, double-angle formulas can be used to rewrite expressions involving double-angle functions in terms of the sine and cosine of the original angle, which may be more convenient for evaluation or further manipulation. By applying these formulas strategically, you can often simplify complex trigonometric expressions into a more manageable form.
Trigonometric identities are mathematical equations that hold true for all values of the variables involved, often used to simplify and manipulate trigonometric expressions.
Angle addition formulas express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine and cosine of the individual angles.