Use the given information to find each of the following.
cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°

Verified step by step guidance

1. Identify the quadrant where the angle \( \theta \) lies. Since \( 180^\circ < \theta < 270^\circ \), \( \theta \) is in the third quadrant, where sine is negative and cosine is also negative.
2. Recall the half-angle formula for cosine: \[ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \]. We need to find \( \cos \theta \) first.
3. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \). Given \( \sin \theta = -\frac{4}{5} \), calculate \( \cos \theta = \pm \sqrt{1 - \sin^2 \theta} = \pm \sqrt{1 - \left(-\frac{4}{5}\right)^2} \).
4. Determine the correct sign of \( \cos \theta \) based on the quadrant. Since \( \theta \) is in the third quadrant, \( \cos \theta \) is negative.
5. Substitute the value of \( \cos \theta \) into the half-angle formula and determine the sign of \( \cos \frac{\theta}{2} \) based on the quadrant where \( \frac{\theta}{2} \) lies. Since \( \theta \) is between 180° and 270°, \( \frac{\theta}{2} \) is between 90° and 135°, which is in the second quadrant where cosine is negative.

Key Concepts

Understanding the Quadrant and Sign of Trigonometric Functions

The quadrant in which an angle lies determines the sign of its sine, cosine, and tangent values. For 180° < θ < 270°, θ is in the third quadrant where sine and cosine are both negative. This information is crucial for correctly determining the sign of trigonometric values when using identities.

Half-Angle Identities

Half-angle identities allow calculation of trigonometric functions of half an angle using the functions of the original angle. For cosine, the identity is cos(θ/2) = ±√[(1 + cos θ)/2]. The sign depends on the quadrant of θ/2, so understanding the angle's location is essential.

Using Pythagorean Identity to Find cos θ

Given sin θ, cos θ can be found using the Pythagorean identity sin²θ + cos²θ = 1. By substituting the known sine value, cos θ is calculated as ±√(1 - sin²θ). The correct sign is chosen based on the quadrant of θ.