Trig Speed Test Practice Quiz

Sin 30° equals 0.5 based on the standard ratios in a 30-60-90 triangle. This fundamental value is essential for quick trigonometric calculations.

Cosine of 60° is 0.5 as derived from the relationships in a 30-60-90 triangle. Recognizing this value is a key step in many trigonometric problems.

Tan 45° equals 1 because both sin(45°) and cos(45°) are equal, making their ratio 1. This basic fact forms the basis for understanding more complex trigonometric relationships.

Cosine of 0° is 1, a fundamental constant in trigonometry. This value is frequently used in solving more advanced problems.

Sine is defined as the ratio of the opposite side to the hypotenuse. Mastering this definition is crucial for solving right-triangle problems efficiently.

Within a 360° rotation, the sine function equals 0.5 at 30° and 150°. This result comes from the symmetry of the sine curve on the unit circle.

The identity sin²(x) + cos²(x) = 1 is the foundational Pythagorean identity in trigonometry. It is extensively used to simplify and manipulate trigonometric expressions.

Tan(x) is defined as the ratio of sine to cosine, meaning tan(x) = sin(x)/cos(x). This relationship is crucial for rewriting and simplifying trigonometric expressions.

Using the Pythagorean identity, sin(x) can be computed as √(1 - cos²(x)) which gives √(1 - 0.36) = 0.8. Since x is in the first quadrant, the sine value is positive.

270° corresponds to 3π/2 radians since 180° is equal to π radians. This conversion is fundamental when working with trigonometric functions in different units.

To convert 150° to radians, multiply by π/180 resulting in 5π/6. This conversion is a basic yet essential skill in trigonometry.

The amplitude represents the maximum deviation from the midline, and for y = 3sin(x), the amplitude is 3. This value is useful for graphing sine functions accurately.

The period of a sine function of the form sin(Bx) is given by 2π/B. For sin(2x), the period is 2π/2 = π. This concept is vital for understanding function transformations.

The reference angle is the acute angle formed by the terminal side of an angle and the horizontal axis. For 210°, subtracting 180° gives a reference angle of 30°.

Cos(330°) equals √3/2 because 330° is the reflection of 30° across the x-axis, and cosine remains positive in the fourth quadrant. This showcases the symmetry of cosine values in the unit circle.

Setting cos(2x) equal to zero means 2x must be an odd multiple of 90°. Dividing each solution by 2 yields x = 45°, 135°, 225°, and 315° in the interval [0°, 360°).

The double angle identity for sine states that sin(2x) = 2 sin(x)cos(x). Rearranging this identity gives sin(x)cos(x) = (1/2) sin(2x).

Factoring the quadratic 2sin²(x) - sin(x) - 1 = 0 yields (2sin(x) + 1)(sin(x) - 1) = 0. The valid solutions in [0°, 360°) are from sin(x) = 1 (x = 90°) and sin(x) = -1/2 (x = 210° and 330°).

Secant is the reciprocal of cosine and since cos(45°) = √2/2, sec(45°) is given by 1/(√2/2) which simplifies to √2. Knowing reciprocal relationships is vital in trigonometry.

The sine function equals zero at all integer multiples of π. Expressing the general solution as x = nπ, where n is any integer, captures all such points.