Unit 4: Trigonometry — Directed Angles, Circular Measure, Trigonometric Functions, Related Angles and Graphs

• A directed angle is generated by a ray rotating about its endpoint from an initial side to a terminal side: counter-clockwise is positive, clockwise is negative.
• The angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis.
• The quadrant of the angle is the quadrant in which its terminal side lies: 0–90° first, 90–180° second, 180–270° third, 270–360° fourth.
• Angles whose sides are parallel to the axes (0°, 90°, 180°, 270°, 360°) are quadrantal and belong to no quadrant.

Coterminal angles and revolutions

• Coterminal angles share the initial and terminal sides and differ by a multiple of 360°: θ + 360°k, k ∈ ℤ.
• Half a turn = 180°, one turn = 360°, two and a quarter turns = 810°; add 360° for every full turn.
• To name the angle from a figure: measure the rotation from the positive x-axis in the given direction, keeping the sign.

• Degree: 1/360 of one revolution; 1° = 60′, 1′ = 60″, so 1° = 3600″.
• Decimal degrees to DMS: keep the integer part as degrees, multiply the fraction by 60 for minutes, then again by 60 for seconds.
• The radian is the angle whose intercepted arc equals the radius; one revolution = 2π radians, hence π rad = 180° and 1 rad ≈ 57.3° ≈ 57°16′.

Conversion

• Degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π.
• Common values: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 120° = 2π/3, 135° = 3π/4, 150° = 5π/6, 210° = 7π/6, 270° = 3π/2, 360° = 2π.
• Never mix units inside one expression — convert everything first, and leave the answer in the unit the question asks for.

• l = rθ with θ in radians — for degrees use l = 2πr × (θ/360).
• Sector area = ½ r² θ (θ in radians), equivalently ½ × l × r.
• Area of the corresponding triangle = ½ r² sinθ, so the segment area = sector − triangle = ½ r²(θ − sinθ).
• Gears, wheels and the movement of clock hands are applications: the linear speed of a point = r × (angular speed in rad/s).

Worked example

• r = 10 cm, θ = 60° = π/3 rad: l = 10π/3 cm and sector = ½ × 100 × π/3 = 50π/3 cm².

• Take a point P(x, y) on the terminal side at distance r = √(x² + y²) from the origin: sinθ = y/r, cosθ = x/r, tanθ = y/x (x ≠ 0).
• Reciprocals: cscθ = r/y, secθ = r/x, cotθ = x/y — csc is the reciprocal of sin, sec of cos, cot of tan.
• tanθ = sinθ/cosθ and cotθ = cosθ/sinθ; cotθ is undefined when sinθ = 0, tanθ and secθ when cosθ = 0.

Signs in the quadrants

• First: all six positive. Second: only sin (and csc) positive. Third: only tan (and cot) positive. Fourth: only cos (and sec) positive.
• Mnemonic: C AST — All, Sin, Tan, Cos going round the quadrants counter-clockwise.

Quadrantal values

• 0°: sin 0, cos 1, tan 0. 90°: sin 1, cos 0, tan undefined. 180°: sin 0, cos −1, tan 0. 270°: sin −1, cos 0, tan undefined. 360°: as 0°.

Special acute angles

• 30°: sin = 1/2, cos = √3/2, tan = 1/√3. 45°: sin = cos = √2/2, tan = 1. 60°: sin = √3/2, cos = 1/2, tan = √3.
• The 3-4-5 approximations used in the syllabus: sin 37° ≈ 0.6, cos 37° ≈ 0.8, tan 37° ≈ 0.75; sin 53° ≈ 0.8, cos 53° ≈ 0.6, tan 53° ≈ 4/3.

The three Pythagorean identities

• sin²θ + cos²θ = 1.
• 1 + tan²θ = sec²θ (divide the first by cos²θ).
• 1 + cot²θ = csc²θ (divide the first by sin²θ).

Method when one function is given

• Determine the quadrant from the sign of the given function and the stated condition, so the signs of the rest are fixed.
• Use sin² + cos² = 1 for the missing leg, then tan = sin/cos, and take reciprocals for csc, sec, cot.
• Geometric alternative: build a right triangle with the given ratio and place it in the correct quadrant to read the signs of x, y and r.
• Remember r > 0 always, so sin and cos carry the sign of y and x respectively.

• The reference angle is the acute angle between the terminal side and the x-axis; the value of the function is that of the reference angle with the sign fixed by the quadrant.

The eight forms

• 180° − θ: sin(180−θ) = sinθ, cos(180−θ) = −cosθ, tan(180−θ) = −tanθ.
• 180° + θ: sin = −sinθ, cos = −cosθ, tan = tanθ.
• 360° − θ: sin = −sinθ, cos = cosθ, tan = −tanθ.
• −θ: sin = −sinθ, cos = cosθ, tan = −tanθ (cosine is even, sine and tangent odd).
• 90° − θ (co-functions): sin = cosθ, cos = sinθ, tan = cotθ.
• 90° + θ: sin = cosθ, cos = −sinθ, tan = −cotθ.
• 270° − θ: sin = −cosθ, cos = −sinθ, tan = cotθ.
• 270° + θ: sin = −cosθ, cos = sinθ, tan = −cotθ.
• Rule of thumb: for 90° and 270° the function changes to its co-function; for 0°, 180° and 360° it stays the same — the sign comes from the quadrant of the whole angle.

• Reduce every term to the same acute angle first, then collect like terms.
• Example: [sin(180+θ) · cos(360−θ)] / [tan(90+θ) · cos(90−θ)] = [(−sinθ)(cosθ)] / [(−cotθ)(sinθ)] = (−sinθ cosθ)/(−cosθ) = sinθ.
• For proving identities, transform the harder side only, one valid step at a time, and never "assume the result then work backwards" without reversing the steps.
• Powers are written sin²θ = (sinθ)²; note that sin²θ + sin²(90−θ) = sin²θ + cos²θ = 1 — a frequent one-mark item.

y = sin x and y = cos x

• Both oscillate between −1 and +1 with period 360° (2π rad); the cosine curve is the sine curve shifted left by 90°.
• Sine passes through (0, 0), peaks at 90°, returns to zero at 180°, minimum at 270°, and completes at 360°.
• Key points to plot: 0°, 90°, 180°, 270°, 360° — five points are enough for one cycle.

y = tan x

• Period 180° (π rad), range all reals, with vertical asymptotes at x = 90° + 180°k where cosine vanishes; the curve crosses zero at 0°, 180°, 360°.
• Between two asymptotes the tangent increases continuously from −∞ to +∞.

Reading graphs

• Solutions of an equation such as sin x = ½ on 0° ≤ x < 360° are the x-coordinates where the curve cuts the line y = ½ — here x = 30° and x = 150°.
• Number of solutions equals the number of intersection points, so always state the interval.

• Amplitude = |a|: the factor a stretches the curve vertically, so the range becomes [−|a|, |a|].
• Period = 360°/|b| in degrees, or 2π/|b| in radians: the factor b compresses or stretches the curve horizontally.
• Example: y = 3 sin 2x has amplitude 3 and period 180°; y = ½ cos 4x has amplitude ½ and period 90°.
• A negative a reflects the curve in the x-axis without changing amplitude or period; a negative b does not change the period either.
• Real-life use: tides, alternating current, and the height of a point on a wheel — read the maximum, minimum and one complete cycle from the graph.

Acute angle first

• Ignore the sign, find the acute reference angle from the special values or the tables, then restore the quadrant.

All solutions on 0° ≤ θ < 360°

• Fix the two quadrants where the function has the given sign, then write θ = reference angle and θ = 180° ± reference or 360° − reference as appropriate.
• Example: cosθ = −½ with 0° ≤ θ < 360°: reference 60°, cosine negative in the second and third quadrants → θ = 120° or θ = 240°.
• In radians the same idea gives, for sinθ = ½ on 0 ≤ θ < 2π: θ = π/6 or θ = 5π/6.

Simple equations

• Isolate the function first: 2sinθ − 1 = 0 → sinθ = ½; never divide the whole equation by the function name.
• For quadratic-type equations such as 2cos²θ − cosθ − 1 = 0, put u = cosθ, factor (2u + 1)(u − 1) = 0 and solve each branch with its own quadrants.
• Reject any value outside the stated interval, and check that |sin|, |cos| ≤ 1 — a ratio like sinθ = 1.5 has no solution.

• Given P(x, y) on the terminal side: r = √(x² + y²), then sinθ = y/r, cosθ = x/r, tanθ = y/x — signs come from the coordinates, not from a diagram assumption.
• Example: P(−3, 4) gives r = 5, so sinθ = 4/5, cosθ = −3/5, tanθ = −4/3 (second quadrant).
• If the angle is asked in radians and the answer must be exact, keep π in the result: 150° = 5π/6.
• For the general coterminal angle write θ + 360°k or θ + 2πk, and restrict k when the question limits the interval.