Chapter 6: Trigonometric Functions

6.1 Positive Angles and Negative Angles

• Positive angles are measured anticlockwise from the positive x-axis; negative angles are measured clockwise. (p.190)
• For an angle greater than 360° (or 2π rad), subtract multiples of 360° (or 2π) to find its quadrant. (p.191)
• Conversions: q° = q° × π/180° rad; q rad = q × 180/π degrees; 60′ = 1°. (p.191)

6.2 Trigonometric Ratios of Any Angle

• cosec θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ. (p.193)
• Complementary angle formulae: sin θ = cos(90° − θ); cos θ = sin(90° − θ); tan θ = cot(90° − θ). (p.194)
• sec θ = cosec(90° − θ); cosec θ = sec(90° − θ); cot θ = tan(90° − θ). (p.194)
• Reference angle α: Quadrant I α = θ; II α = 180° − θ; III α = θ − 180°; IV α = 360° − θ. (p.197)
• Signs (ASTC): Quadrant I all positive; II sin and cosec; III tan and cot; IV cos and sec. (p.197)
• Steps without a calculator: locate the quadrant, decide the sign, find the reference angle, use its ratio. (p.197)
• sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3. (p.199)
• sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1. (p.199)
• sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. (p.199)

6.3 Graphs of Sine, Cosine and Tangent Functions

• y = sin x and y = cos x: amplitude 1 (max 1, min −1), period 360° or 2π. (p.202)
• y = tan x: no maximum or minimum, period 180° or π, with asymptotes at x = 90° and 270°. (p.202)
• y = sin x has x-intercepts at 0, π, 2π and y-intercept 0. (p.202)
• y = cos x has y-intercept 1. (p.202)
• For y = a sin bx + c: a gives the amplitude |a|, b gives the number of cycles (period 360°/b or 2π/b), c is a translation of (0, c). (p.204)
• For y = tan bx the period is 180°/b or π/b. (p.204)
• Draw a trigonometric graph with at least eight points per cycle. (p.204)
• To sketch y = a sin bx + c for 0 ≤ x ≤ nπ: number of classes = b × n × 2; class interval = nπ / classes. (p.206)
• The graph y = |f(x)| reflects the part below the x-axis onto the upper side. (p.206)
• Graphical solution of a trigonometric equation: draw both graphs on the same axes; solutions are the x-values at the points of intersection. (p.207)
• The number of intersection points equals the number of solutions. (p.208)

6.4 Basic Identities

• sin²A + cos²A = 1. (p.212)
• 1 + tan²A = sec²A. (p.212)
• 1 + cot²A = cosec²A. (p.212)
• tan θ = sin θ/cos θ; cot θ = 1/tan θ; cosec θ = 1/sin θ. (p.211)
• To prove identities: start from the more complex side, convert to basic ratios, and multiply by a conjugate if required. (p.213)

6.5 Addition Formulae and Double Angle Formulae

• sin(A ± B) = sin A cos B ± cos A sin B. (p.215)
• cos(A ± B) = cos A cos B ∓ sin A sin B. (p.215)
• tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B). (p.215)
• sin(A + B) ≠ sin A + sin B. (p.215)
• sin 2A = 2 sin A cos A. (p.218)
• cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A. (p.218)
• tan 2A = 2 tan A/(1 − tan²A). (p.218)
• Half-angle formulae: sin(A/2) = ±√[(1 − cos A)/2]; cos(A/2) = ±√[(1 + cos A)/2]; tan(A/2) = ±√[(1 − cos A)/(1 + cos A)]. (p.220)

6.6 Trigonometric Function Applications

• Without a stated range, a trigonometric equation has infinitely many solutions. (p.222)
• Steps to solve: simplify using identities, find the reference angle, find angles in the correct quadrants, write the solutions. (p.222)
• When solving cos 2q = k for 0° ≤ q ≤ 360°, find 2q for 0° ≤ 2q ≤ 720° first. (p.222)
• Use the calculator in radian mode when the range is in radians. (p.223)
• For equations like sin 2x + cos x = 0, use sin 2x = 2 sin x cos x and factorise. (p.223)
• Problems with angles of elevation q, 2q, 3q: use tan 3q = tan(q + 2q). (p.225)