- Identify the given value as \(\tan(\theta) = -8\) and the goal as finding \(\tan(2\theta)\).
- Recall the double angle formula for the tangent function: \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\).
- Substitute \(-8\) into the formula: \(\tan(2\theta) = \frac{2(-8)}{1 - (-8)^2}\).
- Simplify the expression to find the final value: \(\tan(2\theta) = \boxed{\frac{16}{63}}\).
Explanation
- Understanding the Problem and Identifying the Identity
We are given that \(\tan(\theta) = -8\) for \(\theta\) in the second quadrant, where \(\frac{\pi}{2} < \theta < \pi\). We want to determine the exact value of \(\tan(2\theta)\). The double angle identity for the tangent function allows us to express \(\tan(2\theta)\) solely in terms of \(\tan(\theta)\). This formula is:
\[\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\]
This identity is useful because it directly relates our given value to the target value without needing to compute \(\sin(\theta)\) or \(\cos(\theta)\) individually.
- Calculation and Conclusion
Now, we substitute the known value \(\tan(\theta) = -8\) into the double angle identity:
\[\tan(2\theta) = \frac{2(-8)}{1 - (-8)^2}\]
Next, we calculate the numerator and the denominator separately. The numerator is:
\[2 \times (-8) = -16\]
The denominator is:
\[1 - (-8)^2 = 1 - 64 = -63\]
Combining these, we get:
\[\tan(2\theta) = \frac{-16}{-63}\]
Since both the numerator and denominator are negative, the fraction simplifies to a positive value:
\[\tan(2\theta) = \frac{16}{63}\]
Thus, the exact value of \(\tan(2\theta)\) is \(\frac{16}{63}\).
Examples
Trigonometric identities like the double angle formula are essential in various fields such as physics, engineering, and computer graphics. For example, when calculating the trajectory of a projectile, engineers often use double angle identities to simplify expressions involving the angle of launch. Similarly, in game development, these formulas help in calculating rotation and projection of 3D objects onto 2D screens, ensuring smooth and accurate movement. By mastering these identities, you gain a powerful tool for simplifying complex mathematical problems in practical applications.