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Difficulty: EasyFundamental Trigonometric Identities

For an angle θ\theta such that π<θ<3π2\pi < \theta < \frac{3\pi}{2}, the value of cos⁡θ=−0.8\cos\theta = -0.8. What is the value of 10sin⁡θ10\sin\theta?

Answer: -6

Answer

The value of 10sin⁡θ10\sin\theta is −6-6.
The correct answer is −6-6. Substituting cos⁡θ=−0.8\cos\theta = -0.8 into the fundamental Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 yields sin⁡2θ+0.64=1\sin^2\theta + 0.64 = 1, which simplifies to sin⁡2θ=0.36\sin^2\theta = 0.36. Since the angle θ\theta lies in the interval π<θ<3π2\pi < \theta < \frac{3\pi}{2} (Quadrant III), its sine value must be negative. Thus, sin⁡θ=−0.6\sin\theta = -0.6. Multiplying this value by 10 gives 10sin⁡θ=−610\sin\theta = -6.

Step-by-Step Solution

1
Use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 with the given cosine value cos⁡θ=−0.8\cos\theta = -0.8.
sin⁡2θ+(−0.8)2=1⇒sin⁡2θ+0.64=1⇒sin⁡2θ=0.36\sin^2\theta + (-0.8)^2 = 1 \Rightarrow \sin^2\theta + 0.64 = 1 \Rightarrow \sin^2\theta = 0.36
The Pythagorean identity relates the sine and cosine values of any angle.
2
Take the square root of both sides, selecting the correct sign based on the quadrant constraint π<θ<3π2\pi < \theta < \frac{3\pi}{2}.
Since the angle lies in Quadrant III, the sine function must be negative. Thus, sin⁡θ=−0.36=−0.6\sin\theta = -\sqrt{0.36} = -0.6.
In the third quadrant, y-coordinates on the unit circle are negative, meaning sin⁡θ\sin\theta must be negative.
3
Multiply sin⁡θ\sin\theta by 10 to get the final requested value.
10sin⁡θ=10×(−0.6)=−610\sin\theta = 10 \times (-0.6) = -6
This calculation yields the final answer requested by the problem.

Key Concept

Applying the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 and choosing the correct sign based on the angle's quadrant.
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Topics

Fundamental Trigonometric Identities

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