The tangent function of twice an angle, where θ represents the original angle. This term is crucial in understanding double-angle, half-angle, and reduction formulas, which are essential concepts in pre-calculus.
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The formula for $\tan(2\theta)$ is $\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}$.
The double-angle formula for $\tan(2\theta)$ is particularly useful in simplifying trigonometric expressions and solving trigonometric equations.
The half-angle formula for $\tan(\theta/2)$ can be derived from the double-angle formula for $\tan(2\theta)$.
Reduction formulas, such as those for $\sin(180^\circ-\theta)$ and $\cos(180^\circ-\theta)$, can be expressed in terms of $\tan(2\theta)$.
The term $\tan(2\theta)$ is crucial in understanding the relationships between trigonometric functions and in solving complex trigonometric problems.
Review Questions
Explain how the formula for $\tan(2\theta)$ is derived from the basic trigonometric identities.
The formula for $\tan(2\theta)$ can be derived by starting with the double-angle formulas for $\sin(2\theta)$ and $\cos(2\theta)$, which are $\sin(2\theta) = 2\sin\theta\cos\theta$ and $\cos(2\theta) = \cos^2\theta - \sin^2\theta$. Dividing the double-angle formula for $\sin(2\theta)$ by the double-angle formula for $\cos(2\theta)$ and simplifying, we arrive at the formula $\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}$.
Describe how the half-angle formula for $\tan(\theta/2)$ can be derived from the double-angle formula for $\tan(2\theta)$.
The half-angle formula for $\tan(\theta/2)$ can be derived from the double-angle formula for $\tan(2\theta)$ by rearranging the terms. Specifically, the double-angle formula can be rewritten as $\tan(\theta) = \frac{2\tan(\theta/2)}{1-\tan^2(\theta/2)}$. Solving this equation for $\tan(\theta/2)$ gives the half-angle formula: $\tan(\theta/2) = \pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$.
Explain how the term $\tan(2\theta)$ can be used to simplify reduction formulas for trigonometric functions.
The term $\tan(2\theta)$ can be used to simplify reduction formulas, such as those for $\sin(180^\circ-\theta)$ and $\cos(180^\circ-\theta)$. By expressing these reduction formulas in terms of $\tan(2\theta)$, we can take advantage of the properties of the tangent function and its relationship to the other trigonometric functions. This allows for more efficient simplification and manipulation of trigonometric expressions, which is crucial in solving complex pre-calculus problems.