Find sinθ.
tan θ = -(√7)/2, sec θ > 0

Verified step by step guidance

1. Identify the given information: \( \tan \theta = -\frac{\sqrt{7}}{2} \) and \( \sec \theta > 0 \). Recall that \( \sec \theta = \frac{1}{\cos \theta} \), so \( \sec \theta > 0 \) means \( \cos \theta > 0 \).
2. Determine the quadrant where \( \theta \) lies. Since \( \tan \theta \) is negative and \( \cos \theta \) is positive, \( \theta \) must be in the fourth quadrant (where cosine is positive and tangent is negative).
3. Use the identity relating tangent and sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Let \( \cos \theta = x \), then \( \sin \theta = \tan \theta \times x = -\frac{\sqrt{7}}{2} x \).
4. Apply the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = -\frac{\sqrt{7}}{2} x \) and \( \cos \theta = x \) to get \( \left(-\frac{\sqrt{7}}{2} x\right)^2 + x^2 = 1 \).
5. Solve the equation for \( x \) (which is \( \cos \theta \)), then use \( \sin \theta = -\frac{\sqrt{7}}{2} x \) to find \( \sin \theta \). Remember to choose the sign of \( \sin \theta \) consistent with the quadrant (fourth quadrant means \( \sin \theta < 0 \)).

Key Concepts

Trigonometric Ratios and Their Relationships

Trigonometric ratios like sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. Knowing one ratio, such as tangent, allows you to find others using identities or the Pythagorean theorem.

Sign of Trigonometric Functions in Different Quadrants

The signs of sine, cosine, and tangent depend on the quadrant where the angle lies. Given tan θ = -(√7)/2 and sec θ > 0, you can determine the quadrant by recalling that sec θ is positive where cosine is positive.

Using Pythagorean Identities to Find Missing Ratios

Pythagorean identities like 1 + tan²θ = sec²θ help find unknown trigonometric values. By substituting the given tangent value and using the sign information, you can calculate sine θ accurately.