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Difficulty: MediumTrigonometric Ratios and Identities

In right triangle DEFDEF, the measure of angle FF is 90∘90^\circ. If cos⁡(D)=35\cos(D) = \frac{3}{5}, what is the value of sin⁡(E)−tan⁡(D)\sin(E) - \tan(D)?

  1. A
    −815-\frac{8}{15}
  2. B
    −320-\frac{3}{20}
  3. −1115-\frac{11}{15}Answer
  4. D
    2915\frac{29}{15}

Answer

−1115-\frac{11}{15}
To find the value of sin⁡(E)−tan⁡(D)\sin(E) - \tan(D), we first determine the trigonometric ratios for triangle DEFDEF. Since DD and EE are the two acute angles in right triangle DEFDEF with the right angle at FF, the angles are complementary, which means sin⁡(E)=cos⁡(D)=35\sin(E) = \cos(D) = \frac{3}{5}. Next, using the definition of cosine, cos⁡(D)=adjacenthypotenuse=35\cos(D) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}. By the Pythagorean theorem, the length of the opposite side to angle DD is 52−32=4\sqrt{5^2 - 3^2} = 4. Therefore, tan⁡(D)=oppositeadjacent=43\tan(D) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}. Substituting these values into the expression gives sin⁡(E)−tan⁡(D)=35−43=9−2015=−1115\sin(E) - \tan(D) = \frac{3}{5} - \frac{4}{3} = \frac{9 - 20}{15} = -\frac{11}{15}.

Step-by-Step Solution

1
Identify the relationship between the acute angles in right triangle DEFDEF.
Since angle FF is 90∘90^\circ, angles DD and EE are complementary, meaning D+E=90∘D + E = 90^\circ. By the co-function identity, sin⁡(E)=cos⁡(D)\sin(E) = \cos(D).
This allows us to determine sin⁡(E)\sin(E) directly from the given value of cos⁡(D)\cos(D) without finding angle measures.
2
Determine the value of sin⁡(E)\sin(E) and find the side lengths of triangle DEFDEF.
sin⁡(E)=cos⁡(D)=35\sin(E) = \cos(D) = \frac{3}{5}. Using the ratio cos⁡(D)=adjacenthypotenuse=35\cos(D) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}, we let the adjacent side DF=3DF = 3 and the hypotenuse DE=5DE = 5. By the Pythagorean theorem, the opposite side is EF=52−32=4EF = \sqrt{5^2 - 3^2} = 4.
These side lengths are necessary to calculate the value of tan⁡(D)\tan(D).
3
Calculate the value of tan⁡(D)\tan(D) and evaluate the given expression.
tan⁡(D)=oppositeadjacent=EFDF=43\tan(D) = \frac{\text{opposite}}{\text{adjacent}} = \frac{EF}{DF} = \frac{4}{3}. Evaluating the expression: sin⁡(E)−tan⁡(D)=35−43=9−2015=−1115\sin(E) - \tan(D) = \frac{3}{5} - \frac{4}{3} = \frac{9 - 20}{15} = -\frac{11}{15}.
This completes the subtraction to find the final value of the expression.

Key Concept

Trigonometric ratios in right triangles and co-function identities of complementary angles.

Alternative Method

Instead of using the co-function identity, one can draw a right triangle DEFDEF with adjacent side DF=3DF = 3 and hypotenuse DE=5DE = 5. Using the Pythagorean theorem, the opposite side EF=4EF = 4. From the triangle, sin⁡(E)=DFDE=35\sin(E) = \frac{DF}{DE} = \frac{3}{5} and tan⁡(D)=EFDF=43\tan(D) = \frac{EF}{DF} = \frac{4}{3}. Evaluating the expression gives 35−43=−1115\frac{3}{5} - \frac{4}{3} = -\frac{11}{15}.
Estimated Time:1m 30s
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Topics

Trigonometric Ratios and Identities

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