Examkin

Soru

Zorluk: ZorFundamental Trigonometric Identities

For an angle θ\theta such that π<θ<3π2\pi < \theta < \frac{3\pi}{2}, the equation 1−sin⁡θcos⁡θ+cos⁡θ1−sin⁡θ=−103\frac{1 - \sin\theta}{\cos\theta} + \frac{\cos\theta}{1 - \sin\theta} = -\frac{10}{3} is satisfied. What is the value of sin⁡θ\sin\theta?

  1. −45-\frac{4}{5}Cevap
  2. B
    −35-\frac{3}{5}
  3. C
    45\frac{4}{5}
  4. D
    −34-\frac{3}{4}
  5. E
    35\frac{3}{5}

Cevap

The value of sin⁡θ\sin\theta is −45-\frac{4}{5}.
The correct answer is determined by first rewriting the given equation by finding a common denominator, which simplifies the numerator to 2(1−sin⁡θ)2(1-\sin\theta) using the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. The term (1−sin⁡θ)(1-\sin\theta) cancels out, resulting in 2cos⁡θ=−103\frac{2}{\cos\theta} = -\frac{10}{3}, which implies cos⁡θ=−35\cos\theta = -\frac{3}{5}. In Quadrant III, sine is negative, so using the identity sin⁡θ=−1−cos⁡2θ\sin\theta = -\sqrt{1 - \cos^2\theta} yields the correct value.

Adım Adım Çözüm

1
Find a common denominator to add the fractions on the left-hand side of the equation.
The expression becomes (1−sin⁡θ)2+cos⁡2θcos⁡θ(1−sin⁡θ)\frac{(1 - \sin\theta)^2 + \cos^2\theta}{\cos\theta(1 - \sin\theta)}.
To combine the algebraic terms into a single rational expression.
2
Expand the numerator and apply the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1.
The numerator simplifies to 1−2sin⁡θ+sin⁡2θ+cos⁡2θ=1−2sin⁡θ+1=2−2sin⁡θ=2(1−sin⁡θ)1 - 2\sin\theta + \sin^2\theta + \cos^2\theta = 1 - 2\sin\theta + 1 = 2 - 2\sin\theta = 2(1 - \sin\theta).
To reduce the numerator's complexity using fundamental trigonometric identities.
3
Cancel the common factor (1−sin⁡θ)(1 - \sin\theta) from the numerator and denominator, and equate the simplified term to −103-\frac{10}{3}.
The equation simplifies to 2cos⁡θ=−103\frac{2}{\cos\theta} = -\frac{10}{3}, which gives cos⁡θ=−35\cos\theta = -\frac{3}{5}.
To solve for the cosine of the angle θ\theta.
4
Use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 to calculate sin⁡θ\sin\theta, keeping the quadrant sign rules in mind.
Since π<θ<3π2\pi < \theta < \frac{3\pi}{2} (Quadrant III), sin⁡θ\sin\theta is negative. Therefore, sin⁡θ=−1−cos⁡2θ=−1−(−35)2=−45\sin\theta = -\sqrt{1 - \cos^2\theta} = -\sqrt{1 - \left(-\frac{3}{5}\right)^2} = -\frac{4}{5}.
To find the correct value and sign of the sine ratio for the given quadrant.

Anahtar Kavram

Fundamental Trigonometric Identities
Bu soruyu puanla

Konular

Fundamental Trigonometric Identities

Bu konudan daha fazla soru