Mathematics

Mathematics

Problem

For an angle θ\theta such that π2<θ<π\dfrac{\pi}{2}<\theta<\pi and sec⁡θ=−53\sec\theta=-\dfrac{5}{3}, determine the value of sin⁡θ+tan⁡θ\sin\theta+\tan\theta. a) −43-\dfrac{4}{3} b) −815-\dfrac{8}{15} c) 815\dfrac{8}{15} d) 415\dfrac{4}{15}
Intuitive explanation

You're working with a secant value in the second quadrant, where sine is positive but cosine is negative. Start by converting sec⁡θ\sec\theta to cos⁡θ\cos\theta, then use the Pythagorean identity to find sin⁡θ\sin\theta. Once you have both, computing tan⁡θ\tan\theta and adding it to sin⁡θ\sin\theta is straightforward—the key is being careful with signs in the second quadrant.

Intuitive explanation

You're working with a secant value in the second quadrant, where sine is positive but cosine is negative. Start by converting sec⁡θ\sec\theta to cos⁡θ\cos\theta, then use the Pythagorean identity to find sin⁡θ\sin\theta. Once you have both, computing tan⁡θ\tan\theta and adding it to sin⁡θ\sin\theta is straightforward—the key is being careful with signs in the second quadrant.

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