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Difficulty: MediumTrigonometric Ratios and Identities

An acute angle θ\theta in a right triangle satisfies cos⁡(θ)=35\cos(\theta) = \frac{3}{5}. If ϕ\phi is the other acute angle of the triangle, what is the value of sin⁡(ϕ)tan⁡(θ)\sin(\phi) \tan(\theta)?

  1. A
    1615\frac{16}{15}
  2. B
    920\frac{9}{20}
  3. 45\frac{4}{5}Answer
  4. D
    1225\frac{12}{25}

Answer

four-fifths
The correct answer is four-fifths. Because θ\theta and ϕ\phi are complementary angles in a right triangle, sin⁡(ϕ)\sin(\phi) equals cos⁡(θ)\cos(\theta), which is given as three-fifths. Using the Pythagorean identity, sin⁡(θ)\sin(\theta) is four-fifths, making tan⁡(θ)\tan(\theta) equal to four-thirds. Multiplying sin⁡(ϕ)\sin(\phi) and tan⁡(θ)\tan(\theta) gives three-fifths times four-thirds, which simplifies to four-fifths.

Step-by-Step Solution

1
Use the complementary angle relationship to find sin⁡(ϕ)\sin(\phi).
sin⁡(ϕ)=cos⁡(θ)=35\sin(\phi) = \cos(\theta) = \frac{3}{5}
The two acute angles in a right triangle, θ\theta and ϕ\phi, sum to 90∘90^\circ. Therefore, the sine of one angle equals the cosine of the other.
2
Find sin⁡(θ)\sin(\theta) using the Pythagorean identity.
sin⁡(θ)=45\sin(\theta) = \frac{4}{5}
Since sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1, we have sin⁡2(θ)+(35)2=1\sin^2(\theta) + \left(\frac{3}{5}\right)^2 = 1, which gives sin⁡2(θ)=1625\sin^2(\theta) = \frac{16}{25}. Because θ\theta is acute, sin⁡(θ)=45\sin(\theta) = \frac{4}{5}.
3
Calculate tan⁡(θ)\tan(\theta) using the quotient identity.
tan⁡(θ)=43\tan(\theta) = \frac{4}{3}
By definition, tan⁡(θ)=sin⁡(θ)cos⁡(θ)=4/53/5=43\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{4/5}{3/5} = \frac{4}{3}.
4
Multiply sin⁡(ϕ)\sin(\phi) by tan⁡(θ)\tan(\theta).
sin⁡(ϕ)tan⁡(θ)=45\sin(\phi) \tan(\theta) = \frac{4}{5}
Substitute the found values: sin⁡(ϕ)tan⁡(θ)=35×43=45\sin(\phi) \tan(\theta) = \frac{3}{5} \times \frac{4}{3} = \frac{4}{5}.

Key Concept

Trigonometric co-function identities and quotient identities in right triangles
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Topics

Trigonometric Ratios and Identities

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