Determine whether the positive or negative square root should be selected.
cos 58° = ±√ (1 + cos 116°)/2]

Verified step by step guidance

1. Recognize that the expression given is derived from the half-angle identity for cosine: \(\cos \theta = \pm \sqrt{\frac{1 + \cos 2\theta}{2}}\). Here, \(\theta = 58^\circ\) and \(2\theta = 116^\circ\).
2. Understand that the \(\pm\) sign depends on the quadrant in which the angle \(\theta\) lies. Since \(\theta = 58^\circ\) is in the first quadrant (0° to 90°), where cosine values are positive, the positive root should be selected.
3. Confirm the quadrant by recalling that cosine is positive in the first and fourth quadrants and negative in the second and third quadrants.
4. Therefore, for \(\cos 58^\circ\), choose the positive square root in the half-angle formula.
5. Summarize: Use the positive root because \(58^\circ\) is in the first quadrant where cosine is positive.

Key Concepts

Half-Angle Identity for Cosine

The half-angle identity expresses cosine of half an angle in terms of the cosine of the full angle: cos(θ/2) = ±√[(1 + cos θ)/2]. This formula helps find cosine values for angles that are half of known angles, but requires choosing the correct sign based on the angle's quadrant.

Determining the Sign of Trigonometric Functions

The sign of cosine depends on the quadrant in which the angle lies. Cosine is positive in the first and fourth quadrants and negative in the second and third. Identifying the quadrant of the half-angle (here 58°) is essential to select the correct positive or negative root.

Relationship Between Angles and Their Cosine Values

Understanding how cosine values relate for angles and their multiples or halves is crucial. For example, cos 58° relates to cos 116° through the half-angle formula. Recognizing these relationships allows simplification and correct evaluation of trigonometric expressions.