Suppose θ is in the interval (90°, 180°). Find the sign of each of the following. sec(θ + 180°)

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Recall the definition of the secant function: \(\sec(\alpha) = \frac{1}{\cos(\alpha)}\). To determine the sign of \(\sec(\theta + 180^\circ)\), we need to analyze the sign of \(\cos(\theta + 180^\circ)\).

Use the cosine angle addition identity for a shift by \(180^\circ\): \(\cos(\theta + 180^\circ) = -\cos(\theta)\).

Since \(\theta\) is in the interval \((90^\circ, 180^\circ)\), determine the sign of \(\cos(\theta)\) in this interval. Recall that cosine is negative in the second quadrant (between \(90^\circ\) and \(180^\circ\)).

Given that \(\cos(\theta)\) is negative in this interval, substitute back into the expression \(\cos(\theta + 180^\circ) = -\cos(\theta)\) to find its sign. Since \(\cos(\theta)\) is negative, \(-\cos(\theta)\) will be positive.

Finally, since \(\sec(\theta + 180^\circ) = \frac{1}{\cos(\theta + 180^\circ)}\), and \(\cos(\theta + 180^\circ)\) is positive, conclude that \(\sec(\theta + 180^\circ)\) is positive.

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주요 개념

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Trigonometric Function Periodicity

Trigonometric functions repeat their values in regular intervals called periods. For secant, which is the reciprocal of cosine, the period is 360°. This means sec(θ + 360°) = sec(θ), and understanding this helps simplify expressions involving angle shifts.

Angle Addition and Quadrant Analysis

Adding angles shifts the position of the terminal side on the unit circle. Since θ is in (90°, 180°), adding 180° moves the angle to (270°, 360°). Knowing which quadrant the new angle lies in is essential to determine the sign of trigonometric functions.

Sign of Secant Function in Different Quadrants

Secant is the reciprocal of cosine, so its sign depends on the cosine value. Cosine is negative in the second and third quadrants and positive in the first and fourth. Therefore, secant is negative where cosine is negative and positive where cosine is positive.