Trigonometric Identities and Equations

The value of sin 330° can be found using the unit circle. 330° is in the fourth quadrant, where sine is negative. The reference angle is 30° (since 360° - 330° = 30°). sin 30° = 1/2 = 0.5, so sin 330° = -0.5. The other options are incorrect - 0.6 is not a standard sine value, -1.5 is impossible (sine ranges from -1 to 1), and -1.0 would be sin 270°.

The Pythagorean identity states that sin²x + cos²x = 1 for any angle x. This is one of the fundamental identities in trigonometry derived from the unit circle definition. Options 0, 90, and -1 are incorrect values for this expression.

Cosine is an even function, meaning cos(-x) = cos x for all x. This symmetry property holds because cosine is based on the x-coordinate on the unit circle, which is unchanged by reflecting across the x-axis. Sine is odd (sin(-x) = -sin x).

The cosine addition formula is cos(x+y) = cos x cos y - sin x sin y. This is a fundamental trigonometric identity for expanding cosine of a sum. Option A has the wrong sign, B mixes sine and cosine incorrectly, and D just reorders the terms but still has the wrong sign.

The double-angle formula for sine is sin 2x = 2 sin x cos x. This identity can be derived from the sine addition formula sin(x+x) = sin x cos x + cos x sin x. Option B is missing the factor of 2, and the other options are incorrect.

sin(180° - x) = sin x due to the supplementary angle identity. In the second quadrant (angles 90° to 180°), sine is positive and has the same value as its supplement. This is related to the symmetry of the sine function.

Sine is an odd function, meaning sin(-x) = -sin(x). The graph of sine is symmetric about the origin, so the negative angle produces the negative of the original value. This is a fundamental trigonometric identity.

The Maclaurin series expansion of sin(x) contains only odd powers of x with alternating signs. Option A shows this exact pattern: x - x³/3! + x⁵/5! - ... Option B is the expansion for cos(x) (even powers only). Option C is the expansion for eˣ (all positive terms). Option D is a simple geometric series.

As x goes from π/4 to 3π/4, 2x goes from π/2 to 3π/2. In this range, |sin(2x)| starts at sin(π/2)=1, decreases to sin(π)=0 at x=π/2, then increases to sin(3π/2)=1. So it decreases then increases.

For any angle A, sin²(A) + cos²(A) = 1, so cos(A) = √(1 - sin²(A)). Given sin(A) = 11/14, we have cos(A) = √(1 - 121/196) = √(75/196) = √75/14 = 5√3/14. The cosine is positive since A is acute.

sin(π/6) = 1/2 = 0.5. Option D '.5' is the decimal representation. π/6 radians equals 30 degrees, and sin 30° = 0.5.

The derivative of sin x is cos x (d/dx[sin x] = cos x). This is a fundamental derivative rule from calculus.

sin(0) = 0, so cos(sin 0) = cos(0) = 1. Options A, B, and D are incorrect values. This question tests understanding of trigonometric function composition and the values of sin and cos at 0 radians.