Find sinθ.
sec θ = 7/2, tan θ < 0

Verified step by step guidance

1. Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). Given \(\sec \theta = \frac{7}{2}\), find \(\cos \theta\) by taking the reciprocal: \(\cos \theta = \frac{2}{7}\).
2. Use the Pythagorean identity to find \(\sin \theta\): \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = \frac{2}{7}\) to get \(\sin^2 \theta = 1 - \left(\frac{2}{7}\right)^2\).
3. Simplify the expression for \(\sin^2 \theta\): \(\sin^2 \theta = 1 - \frac{4}{49} = \frac{49}{49} - \frac{4}{49} = \frac{45}{49}\).
4. Take the square root to find \(\sin \theta\): \(\sin \theta = \pm \sqrt{\frac{45}{49}} = \pm \frac{\sqrt{45}}{7}\). Simplify \(\sqrt{45}\) if desired.
5. Determine the correct sign of \(\sin \theta\) using the information \(\tan \theta < 0\). Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\cos \theta\) is positive, \(\sin \theta\) must be negative to make \(\tan \theta\) negative.

Key Concepts

Reciprocal Trigonometric Functions

The secant function (sec θ) is the reciprocal of the cosine function, meaning sec θ = 1/cos θ. Knowing sec θ allows you to find cos θ by taking its reciprocal, which is essential for determining sin θ using trigonometric identities.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. Once cos θ is known, this identity helps calculate sin θ by rearranging to sin θ = ±√(1 - cos²θ). The sign depends on the quadrant where θ lies.

Sign of Trigonometric Functions by Quadrant

The sign of sine, cosine, and tangent functions depends on the quadrant of the angle θ. Given tan θ < 0 and sec θ > 0, θ lies in a quadrant where cosine is positive and tangent is negative, which helps determine the correct sign of sin θ.