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Difficulty: MediumTrigonometric Ratios and Identities

An acute angle θ\theta in a right triangle satisfies the equation cos⁡(θ)=513\cos(\theta) = \frac{5}{13}. What is the value of tan⁡(90∘−θ)\tan(90^\circ - \theta)?

  1. A
    125\frac{12}{5}
  2. B
    1213\frac{12}{13}
  3. 512\frac{5}{12}Answer
  4. D
    513\frac{5}{13}

Answer

512\frac{5}{12}
The value 512\frac{5}{12} is correct because in any right triangle with acute angle θ\theta, the complementary angle is 90∘−θ90^\circ - \theta. The side adjacent to θ\theta becomes the side opposite to 90∘−θ90^\circ - \theta, and the side opposite to θ\theta becomes the side adjacent to 90∘−θ90^\circ - \theta. Given cos⁡(θ)=513\cos(\theta) = \frac{5}{13}, the adjacent side is 55 and the hypotenuse is 1313. The remaining side is 1212 by the Pythagorean theorem. Therefore, tan⁡(90∘−θ)=oppositeadjacent=512\tan(90^\circ - \theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}.

Step-by-Step Solution

1
Identify the relationship between the acute angles in a right triangle.
The two acute angles of a right triangle sum to 90∘90^\circ. Therefore, the other acute angle is 90∘−θ90^\circ - \theta.
Since the sum of angles in any triangle is 180∘180^\circ and one angle is 90∘90^\circ, the remaining two angles must sum to 90∘90^\circ.
2
Determine the side lengths of the right triangle based on the given ratio.
Since cos⁡(θ)=adjacenthypotenuse=513\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{13}, we can let the adjacent side to θ\theta be 55 and the hypotenuse be 1313. Using the Pythagorean theorem, the opposite side to θ\theta is 132−52=169−25=144=12\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12.
The Pythagorean theorem relates the sides of a right triangle: opposite2+adjacent2=hypotenuse2\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2.
3
Calculate the value of tan⁡(90∘−θ)\tan(90^\circ - \theta) using the side lengths.
For the angle (90∘−θ)(90^\circ - \theta), the opposite side is the side adjacent to θ\theta (length 55), and the adjacent side is the side opposite to θ\theta (length 1212). Thus, tan⁡(90∘−θ)=oppositeadjacent=512\tan(90^\circ - \theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}.
By definition, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

Key Concept

Complementary angle trigonometric relationships and Pythagorean triple side determinations in right triangles.

Alternative Method

Alternatively, we can use trigonometric identities: tan⁡(90∘−θ)=cot⁡(θ)=cos⁡(θ)sin⁡(θ)\tan(90^\circ - \theta) = \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}. Since sin⁡(θ)=1−cos⁡2(θ)=1−(513)2=1213\sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}, we get cot⁡(θ)=5/1312/13=512\cot(\theta) = \frac{5/13}{12/13} = \frac{5}{12}.
Estimated Time:1m 15s
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Topics

Trigonometric Ratios and Identities

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