91Ó°ÊÓ

The tangent function is one of the basic trigonometric functions, fundamental to understanding trigonometric equations. In a right triangle, the tangent of an angle \( \theta \) represents the ratio of the opposite side to the adjacent side. This can be written as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

However, tangent can also be defined using the sine and cosine functions as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Tangent is unique because unlike sine and cosine, its value can potentially range from negative infinity to positive infinity. This is critical in solving trigonometric equations.
In this problem, the solution revolves around expressing \( \tan^2 \theta - \tan \theta - 1 = 0 \) as a quadratic equation. Here, \( \tan \theta \) acts as the variable, allowing us to use familiar algebraic techniques to solve it.

The quadratic formula is a powerful tool used in algebra to find the roots of a quadratic equation. A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \). In our trigonometric problem, the equation \( \tan^2 \theta - \tan \theta - 1 = 0 \) is treated as a quadratic with \( x = \tan \theta \). The coefficients are:

• \( a = 1 \)
• \( b = -1 \)
• \( c = -1 \)

We substitute these into the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This gives:

• \( x = \frac{1 \pm \sqrt{5}}{2} \)

After calculating, we obtain two possible values for \( \tan \theta \): approximately 1.618 and -0.618. These are the solutions that arise from the quadratic formula application, essential in identifying potential angles for \( \theta \).

Finding angles that match specific tangent values involves understanding angle measures. Here, we seek angles within the interval \( 0^\circ \leq \theta \leq 360^\circ \) where \( \tan \theta \) is approximately 1.618 and -0.618.
Using an inverse tangent function on calculators reveals:

• For \( \tan \theta = 1.618 \), \( \theta \approx 58.0^\circ \).
• For \( \tan \theta = -0.618 \), \( \theta \approx 328.0^\circ \).

These angle measurements guide us to solve for \( \theta \). It's crucial to remember that the tangent function can be positive or negative in different quadrants, affecting our calculated angles and necessitating the inclusion of complementary angles for a complete solution.

The periodicity of the tangent function means it repeats its values over regular intervals. Specifically, the tangent function has a period of \( 180^\circ \). This property is crucial when finding all solutions for our trigonometric equation within the interval \( 0^\circ \leq \theta \leq 360^\circ \).
Starting with \( \theta = 58.0^\circ \) for \( \tan \theta = 1.618 \), adding the period gives:

• Another angle: \( 238.0^\circ \).

Similarly, for \( \tan \theta = -0.618 \) starting with \( \theta = 328.0^\circ \), subtracting the period results in:

• An additional angle: \( 148.0^\circ \).

These angles reflect the multiple intersections of the tangent graph with specific values throughout its periods and are essential for capturing all possible solutions in the given range.