If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .

Verified step by step guidance

1. insert step 1> Identify the given values: \( \cos x = -0.750 \) and \( \sin x \approx 0.6614 \).
2. insert step 2> Use the identity for tangent of half-angle: \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \).
3. insert step 3> Substitute the given values into the half-angle identity: \( \tan \frac{x}{2} = \frac{1 - (-0.750)}{0.6614} \).
4. insert step 4> Simplify the expression: \( \tan \frac{x}{2} = \frac{1 + 0.750}{0.6614} \).
5. insert step 5> Calculate the expression to find \( \tan \frac{x}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In this context, cosine (cos) and sine (sin) values are given, which are essential for determining the tangent of half the angle (tan x/2). Understanding these functions is crucial for solving problems involving angles and their relationships.

Half-Angle Formulas

Half-angle formulas are trigonometric identities that express the sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the original angle. For tangent, the formula is tan(x/2) = sin(x)/(1 + cos(x)). This concept is vital for calculating tan x/2 when cos x and sin x are known.

Quadrants and Angle Signs

The unit circle is divided into four quadrants, each with specific signs for sine, cosine, and tangent. Knowing the signs of these functions based on the quadrant in which the angle lies is essential for determining the correct values. In this case, since cos x is negative and sin x is positive, x must be in the second quadrant, affecting the calculation of tan x/2.