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Difficulty: MediumFundamental Trigonometric Identities

For an angle θ\theta satisfying π<θ<3π2\pi < \theta < \frac{3\pi}{2}, if cos⁡θ=−1213\cos \theta = -\frac{12}{13}, what is the value of sec⁡θ−cos⁡θtan⁡θ\frac{\sec \theta - \cos \theta}{\tan \theta}?

  1. A
    −1213-\frac{12}{13}
  2. −513-\frac{5}{13}Answer
  3. C
    513\frac{5}{13}
  4. D
    1213\frac{12}{13}
  5. E
    −25169-\frac{25}{169}

Answer

The expression simplifies to sin⁡θ\sin \theta, which equals −513-\frac{5}{13}.
Using identity substitutions sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, 1−cos⁡2θ=sin⁡2θ1 - \cos^2 \theta = \sin^2 \theta, and tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, the expression simplifies directly to sin⁡θ\sin \theta. Since cos⁡θ=−1213\cos \theta = -\frac{12}{13} in Quadrant III where sine is negative, sin⁡θ=−1−(−1213)2=−513\sin \theta = -\sqrt{1 - \left(-\frac{12}{13}\right)^2} = -\frac{5}{13}.

Step-by-Step Solution

1
Simplify the given trigonometric expression using fundamental identities.
sec⁡θ−cos⁡θtan⁡θ=1cos⁡θ−cos⁡θsin⁡θcos⁡θ=1−cos⁡2θcos⁡θsin⁡θcos⁡θ=sin⁡2θsin⁡θ=sin⁡θ\frac{\sec \theta - \cos \theta}{\tan \theta} = \frac{\frac{1}{\cos \theta} - \cos \theta}{\frac{\sin \theta}{\cos \theta}} = \frac{\frac{1 - \cos^2 \theta}{\cos \theta}}{\frac{\sin \theta}{\cos \theta}} = \frac{\sin^2 \theta}{\sin \theta} = \sin \theta
Applying the reciprocal identity sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, the quotient identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, and the Pythagorean identity 1−cos⁡2θ=sin⁡2θ1 - \cos^2 \theta = \sin^2 \theta simplifies the expression directly to sin⁡θ\sin \theta.
2
Calculate the magnitude of sin⁡θ\sin \theta using the Pythagorean identity.
∣sin⁡θ∣=1−cos⁡2θ=1−(−1213)2=1−144169=25169=513|\sin \theta| = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(-\frac{12}{13}\right)^2} = \sqrt{1 - \frac{144}{169}} = \sqrt{\frac{25}{169}} = \frac{5}{13}
The Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 allows finding the absolute value of sin⁡θ\sin \theta.
3
Determine the correct sign for sin⁡θ\sin \theta based on the given quadrant.
sin⁡θ=−513\sin \theta = -\frac{5}{13}
Since π<θ<3π2\pi < \theta < \frac{3\pi}{2}, the angle lies in Quadrant III, where the sine function is negative.

Key Concept

Simplifying expressions using fundamental Pythagorean, reciprocal, and quotient identities while applying quadrant sign rules.
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Topics

Fundamental Trigonometric Identities

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