Question 6 of 7

For an angle with measure alpha in a right triangle, sine of alpha = 5 over 13 and tangent of alpha = 5 over 12. What is the value of cosine of alpha?

\frac{12}{13}

\frac{12}{\sqrt{194}}

\frac{12}{\sqrt{119}}

\frac{13}{12}

Explanation:

Correct. For any right triangle with angle measure $α$ , the three basic trigonometric ratios are defined in terms of the length of the side adjacent (adj) to $α$ , the length of the side opposite (opp) $α$ , and the length of the hypotenuse (hyp): $\sin α = \frac{opp}{hyp}$ , $\cos α = \frac{adj}{hyp}$ , $\tan α = \frac{opp}{adj}$ . Since $\sin α = \frac{5}{13} = \frac{opp}{hyp}$ and $\tan α = \frac{5}{12} = \frac{opp}{adj}$ , one possible triangle will have sides with lengths of $5$ , $12$ , and $13$ as the opposite, adjacent, and hypotenuse sides, respectively. Using the given triangle, $\cos α = \frac{adj}{hyp} = \frac{12}{13}$ .

$The image is a triangle with labels given for one angle and all three sides.$ The image is a triangle with the bottom left angle labeled alpha. The sides are labeled, clockwise from the top, with the numbers 13, 5, 12. The sides are also labeled, clockwise from the top, with the words hypotenuse, opposite, adjacent.

Incorrect. For any right triangle with angle measure $α$ , the three basic trigonometric ratios are defined in terms of the length of the side adjacent (adj) to $α$ , the length of the side opposite (opp) $α$ , and the length of the hypotenuse (hyp): $\sin α = \frac{opp}{hyp}$ , $\cos α = \frac{adj}{hyp}$ , $\tan α = \frac{opp}{adj}$ . Try identifying the possible side lengths of a triangle with the given information. You may have incorrectly attempted to use the Pythagorean theorem to calculate the length of the hypotenuse: $5^{2} + 13^{2} = {(hyp)}^{2} \to hyp = \sqrt{5^{2} + 13^{2}} = \sqrt{194}$ , so $\cos α = \frac{adj}{hyp} = \frac{12}{\sqrt{194}}$ .

Incorrect. For any right triangle with angle measure $α$ , the three basic trigonometric ratios are defined in terms of the length of the side adjacent (adj) to $α$ , the length of the side opposite (opp) $α$ , and the length of the hypotenuse (hyp): $\sin α = \frac{opp}{hyp}$ , $\cos α = \frac{adj}{hyp}$ , $\tan α = \frac{opp}{adj}$ . Try identifying possible side lengths of a triangle with the given information. You may have incorrectly attempted to use the Pythagorean theorem to calculate the length of the hypotenuse: $5^{2} + {(hyp)}^{2} = 12^{2} \to hyp = \sqrt{12^{2} - 5^{2}} = \sqrt{119}$ , so $\cos α = \frac{adj}{hyp} = \frac{12}{\sqrt{119}}$ .

Incorrect. For any right triangle with angle measure $α$ , the three basic trigonometric ratios are defined in terms of the length of the side adjacent (adj) to $α$ , the length of the side opposite (opp) $α$ , and the length of the hypotenuse (hyp): $\sin α = \frac{opp}{hyp}$ , $\cos α = \frac{adj}{hyp}$ , $\tan α = \frac{opp}{adj}$ . You may have thought $\cos α = \frac{hyp}{adj} = \frac{13}{12}$ .

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