Chapter 2 Trigonometry

2.1 Angles in Standard Position In trigonometry, angles are interpreted as rotations of a ray. The starting position of a ray, along the x-axis, is the initial arm of the angle. The final position, after a rotation about the origin, is the terminal arm of the angle. An angle is said to be in standard position, is when an angle’s initial arm is on the positive x-axis and its vertex is at the origin.

Reference Angles A reference angle (θR) is the acute angles whose vertex is the origin and whose arms are the terminal arm of the angle and the x-axis. The reference angle is always positive and measures between 0° and 90°. -The reference angle, θR, is illustrated for angles, θ, in standard position where 0° ≤ θ ≤ 360°. Example: The angles in standard position with a reference angle of 20° are 20°, 160° (180°- 20°), 200° (180° + 20°), and 340° (360°- 20°).

2.2 Trigonometric Ratios of Any Angle Let P (x, y) represent any point in the first quadrant on a circle with radius r. Then P is on the terminal arm of an angle θ in standard position. Then, by the Pythagorean Theorem, r = √ x2 + y2 There are three primary trigonometric ratios in terms of x, y, and r: sin θ = opposite cos θ = adjacent tan θ = opposite hypotenuse hypotenuse adjacent sin θ = y cos θ = x tan θ = y x r x

Determining the Sign of Trigonometric Ratios One way to memorize each quadrant is: Quadrant I: (All ratios are positive) A = All Quadrant II: (Sine ratios are positive) S = Students Quadrant III: (Tan ratios are positive) T = Take Quadrant IV: (Cosine ratios are positive) C = Calculus

Example: The point P(-8, 15) lies on the terminal arm of an angle, θ, in standard position. Determine the exact trigonometric ratios for sin θ, cos θ, and tan θ. r = √ x2 + y2 r = √(-8)2 + (15)2 r = √289 r = 17 sin θ = y cos θ = x tan θ = y r r x sin θ = 15 cos θ = - 8 tan θ = - 15 17 17 8

Recognizing Patterns of Trigonometric Values from 0° to 90° Degree sin θ cos θ tan θ 0° √ 0 = 0 2 √ 4 = 1 30° √1 = 1 2 2 √3 3 45° √2 1 60° 90° √4 = 1 √0 = 0 undefined

2.3 The Sine Law sin A = sin B = sin C a b c The Sine Law is a relationship between the sides and angles in any triangle. Let ΔABC be any triangle where a, b, c, represent the measures of the sides opposite ∠A, ∠B, and ∠C. To use the Sine Law, we need an angle and the side opposite the angle. You also need another angle or side. The formula is: a = b = c sin A sin B sin C or sin A = sin B = sin C a b c

2.4 The Cosine Law The Cosine Law describes the relationship between the cosine of an angle and the lengths of the three sides of an triangle. For any ΔABC, where a, b, and c are the lengths of the sides opposite to ∠A, ∠B, and ∠C, then the formula is: c2 = a2 + b2 - 2ab cos C or a2 = b2 + c2 - 2bc cos A b2 = a2 + c2 - 2ac cos B In order to find the angle, we can rearrange the formula from the top to this formula: cos C = a2 + b2 - c2 or cos A = b2 + c2 - a2 or cos B = a2 + c2 - b2 2ab 2bc 2ac