ch.1 quiz alg and trig

Standard Position

An angle with the vertex at the origin, initial side on the positive x-axis, and a terminal side determined by the angle.

Initial Side

The ray along the positive x-axis from the origin in standard position.

Terminal Side

The ray where the angle ends after rotation from the initial side.

Radians

A measure of angle where 2π radians = 360°, and π radians = 180°; arc length is proportional to the angle measure.

Degrees

A measure of angle where 360° = 2π radians; the circle is divided into 360 equal parts.

Reference Angle (θ′)

The acute angle formed between the terminal side of θ and the x-axis; used to compute trig values regardless of quadrant.

Reference Angle in Quadrant II

If θ is in Quadrant II, the reference angle θ′ = 180° − θ (or π − θ).

Reference Angle in Quadrant III

If θ is in Quadrant III, the reference angle θ′ = θ − 180° (or θ − π).

Reference Angle in Quadrant IV

If θ is in Quadrant IV, the reference angle θ′ = 360° − θ (or 2π − θ).

Coterminal Angles

Angles that share the same terminal side; their measures differ by full rotations (360° or 2π).

P = (x, y) and r

A point P with coordinates (x, y); r is the distance from the origin to P, r = √(x^2 + y^2).

Unit Circle

Circle of radius 1 centered at the origin; x^2 + y^2 = 1; cos θ = x, sin θ = y.

Cosine, Sine, Tangent (ratios)

cos θ = adjacent/hypotenuse; sin θ = opposite/hypotenuse; tan θ = opposite/adjacent.

Reciprocal Trigonometric Identities

sec θ = hypotenuse/adjacent = r/x; csc θ = hypotenuse/opposite = r/y; cot θ = adjacent/opposite = x/y.

Cosine/Sine/Tangent in terms of x, y, r

cos θ = x/r; sin θ = y/r; tan θ = y/x (x ≠ 0).

Unit Circle Coordinates for Standard Angles

On the unit circle, θ = 0°: P = (1,0); θ = 90°: P = (0,1); θ = 180°: P = (−1,0); θ = 270°: P = (0,−1).

Special Right Triangles

30-60-90 triangle with sides in ratio 1:√3:2; 45-45-90 triangle with sides in ratio 1:1:√2; used for exact trig values.

30-60-90 Values

For 30°: sin = 1/2, cos = √3/2, tan = 1/√3; for 60°: sin = √3/2, cos = 1/2, tan = √3.

45-45-90 Values

For 45°: sin = cos = √2/2, tan = 1.

Special Angles on the Unit Circle

Angles with axis coordinates: 0°, 90°, 180°, 270° and their corresponding P points on the unit circle.

cos(-θ)

Cosine is even; cos(-θ) = cos θ.

sec(-θ)

Secant is even; sec(-θ) = sec θ.

sin(-θ)

Sine is odd; sin(-θ) = -sin θ.

csc(-θ)

Cosecant is odd; csc(-θ) = -csc θ.

tan(-θ)

Tangent is odd; tan(-θ) = -tan θ.

cot(-θ)

Cotangent is odd; cot(-θ) = -cot θ.

cos(90° − θ)

Cosine of a complementary angle equals sine: cos(90°−θ) = sin θ.

sec(90° − θ)

Secant of a complementary angle equals cosecant: sec(90°−θ) = csc θ.

sin(90° − θ)

Sine of a complementary angle equals cosine: sin(90°−θ) = cos θ.

csc(90° − θ)

Cosecant of a complementary angle equals secant: csc(90°−θ) = sec θ.

tan(90° − θ)

Tangent of a complementary angle equals cotangent: tan(90°−θ) = cot θ.

cot(90° − θ)

Cotangent of a complementary angle equals tangent: cot(90°−θ) = tan θ.

cos(π/2 − θ)

Cosine of a complementary angle (radians) equals sine: cos(π/2−θ) = sin θ.

sec(π/2 − θ)

Secant of a complementary angle equals cosecant: sec(π/2−θ) = csc θ.

sin(π/2 − θ)

Sine of a complementary angle equals cosine: sin(π/2−θ) = cos θ.

csc(π/2 − θ)

Cosecant of a complementary angle equals secant: csc(π/2−θ) = sec θ.

tan(π/2 − θ)

Tangent of a complementary angle equals cotangent: tan(π/2−θ) = cot θ.

cot(π/2 − θ)

Cotangent of a complementary angle equals tangent: cot(π/2−θ) = tan θ.

tan θ = sin θ / cos θ

Tangent equals sine divided by cosine.

cot θ = cos θ / sin θ

Cotangent equals cosine divided by sine.

sin^2 θ + cos^2 θ = 1

Pythagorean identity: sine squared plus cosine squared equals 1.

1 + tan^2 θ = sec^2 θ

Pythagorean identity relating tan and sec.

1 + cot^2 θ = csc^2 θ

Pythagorean identity relating cot and csc.

csc θ = 1 / sin θ

Cosecant equals reciprocal of sine.

sec θ = 1 / cos θ

Secant equals reciprocal of cosine.

cot θ = 1 / tan θ

Cotangent equals reciprocal of tangent.

sin θ = 1 / csc θ

Sine equals reciprocal of cosecant.

cos θ = 1 / sec θ

Cosine equals reciprocal of secant.

tan θ = 1 / cot θ

Tangent equals reciprocal of cotangent.

Vertex

The common point where two rays meet to form an angle.

Initial side

One ray that forms the starting side of an angle.

Terminal side

The ray that forms the ending side of an angle.

Angle

The figure formed by two rays with a common vertex; it has an initial side and a terminal side.

Counterclockwise

The positive angular direction; angles are measured in this direction, opposite of a clock's direction.

Positive vs negative angles

Angles measured counterclockwise are positive; clockwise angles are negative (indicate direction, not magnitude).

Acute angle

An angle smaller than 90 degrees.

Right angle

An angle exactly equal to 90 degrees.

Obtuse angle

An angle between 90 and 180 degrees.

Straight angle

An angle equal to 180 degrees.

Complementary angles

Two positive angles whose measures sum to 90 degrees; in right triangles the non-right angles are complementary.

Supplementary angles

Two positive angles whose measures sum to 180 degrees.

Complementary functions

Sine and cosine are related through complementary angles (e.g., sin(θ) = cos(90°−θ) in degrees; sin(θ) = cos(π/2−θ) in radians).

Radian

A non-dimensional unit for measuring angles; defined so that the arc length equals the radius. One radian is the angle subtended by an arc length equal to the circle's radius.

Degree-to-radian conversion factor

To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π; keep answers in terms of π (don’t convert π to 3.14).

Unit circle

A circle with radius 1 used to study special angles; the unit circle focuses on these special angles.

Central angle

An angle whose vertex is at the center of a circle.

Degree-to-radian conversion examples

Examples of converting degree measures to radians: 120° → 2π/3; -310° → -31π/18; 720° → 4π; 450° → 5π/2.