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Difficulty: EasyFundamental Trigonometric Identities

Given that cos⁡θ=45\cos\theta = \frac{4}{5} and the terminal side of angle θ\theta lies in Quadrant IV, what is the value of tan⁡θ\tan\theta?

  1. A
    −43-\frac{4}{3}
  2. −34-\frac{3}{4}Answer
  3. C
    −35-\frac{3}{5}
  4. D
    34\frac{3}{4}
  5. E
    43\frac{4}{3}

Answer

−34-\frac{3}{4}
The correct answer is −34-\frac{3}{4}. Since the angle θ\theta has its terminal side in Quadrant IV, its cosine is positive and its sine is negative. Using the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, we find sin⁡2θ+(45)2=1\sin^2\theta + \left(\frac{4}{5}\right)^2 = 1, which simplifies to sin⁡2θ=925\sin^2\theta = \frac{9}{25}. Because sine is negative in Quadrant IV, sin⁡θ=−35\sin\theta = -\frac{3}{5}. Finally, applying the quotient identity tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, we get tan⁡θ=−3/54/5=−34\tan\theta = \frac{-3/5}{4/5} = -\frac{3}{4}.

Step-by-Step Solution

1
Determine the sign of sin⁡θ\sin\theta in Quadrant IV.
sin⁡θ<0\sin\theta < 0
In Quadrant IV, the x-coordinates (representing cosine) are positive, and the y-coordinates (representing sine) are negative.
2
Use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 to calculate the value of sin⁡θ\sin\theta.
sin⁡θ=−35\sin\theta = -\frac{3}{5}
Substituting cos⁡θ=45\cos\theta = \frac{4}{5} gives sin⁡2θ+(45)2=1  ⟹  sin⁡2θ=1−1625=925\sin^2\theta + \left(\frac{4}{5}\right)^2 = 1 \implies \sin^2\theta = 1 - \frac{16}{25} = \frac{9}{25}. Taking the negative square root because sine is negative in Quadrant IV yields −35-\frac{3}{5}.
3
Use the quotient identity tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta} to calculate tan⁡θ\tan\theta.
tan⁡θ=−34\tan\theta = -\frac{3}{4}
Dividing the value of sin⁡θ\sin\theta by cos⁡θ\cos\theta yields −3/54/5=−34\frac{-3/5}{4/5} = -\frac{3}{4}.

Key Concept

Fundamental Trigonometric Identities
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Topics

Fundamental Trigonometric Identities

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