CELE Engineering Mathematics — Plane and Spherical TrigonometryRevision Notes

Formulas

Exam tips

• For identity-verification board items, work one side only — never move terms across the equal sign.
• When a board item asks for sin 2θ and gives a right triangle, use sin 2θ = 2 sin θ cos θ directly from the triangle sides — no need to find the angle.
• Memorize the exact values table: sin/cos/tan at 0°, 30°, 45°, 60°, 90° — these appear constantly as given or intermediate values.
• The ASTC rule speeds up sign assignment in multi-quadrant problems: practice it until automatic.
• On the calculator, set MODE to DEGREES unless the problem explicitly states radians.

Key points

• The six trig functions are defined from a right triangle: SOH-CAH-TOA (sin = opp/hyp, cos = adj/hyp, tan = opp/adj) and their reciprocals csc, sec, cot.
• All six functions can be expressed in terms of sin and cos alone — mastering this simplifies identity proofs.
• The three Pythagorean identities (sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ) must be memorized cold.
• Sum-and-difference formulas are the parent of all multiple-angle identities — derive double-angle by setting A = B.
• Product-to-sum and sum-to-product identities appear less frequently but are useful for integration-related problems.
• Quadrant signs follow the mnemonic ASTC (All Students Take Calculus): All positive in Q1, Sin in Q2, Tan in Q3, Cos in Q4.
• Reference angle: the acute angle formed with the x-axis; used to evaluate any trig function at any angle.
• Cofunctions: sin θ = cos(90° − θ), tan θ = cot(90° − θ), sec θ = csc(90° − θ).

Definitions

Common mistakes

• Writing cos(A + B) = cosA + cosB — this is WRONG. Always expand using the sum formula.
• Forgetting the sign flip in cos(A + B): it uses MINUS, while cos(A − B) uses PLUS.
• Using sin 2θ = 2 sin θ (omitting the cosθ factor) — a very common careless error.
• Confusing sec with sin (sec = 1/cos, not 1/sin — that is csc).
• Not checking the quadrant when applying the Pythagorean identity to find cos θ from sin θ; the sign of cos θ depends on the quadrant.

Formulas

Exam tips

• Always draw a labeled diagram before writing any equation — this alone eliminates most setup errors.
• For two-observation problems, label the two horizontal distances as x and (x + d) and write two tan equations, then solve simultaneously.
• In bearing/navigation problems, convert all bearings to standard angle from East (counterclockwise) before computing vector components.
• Check: the side opposite the larger angle must be the longer side — use this as a quick sanity check.

Key points

• Right triangle problems form the simplest class: one angle is 90°, so only two trig ratios are needed.
• Angle of elevation: measured upward from the horizontal to the line of sight toward an object above.
• Angle of depression: measured downward from the horizontal to the line of sight toward an object below.
• For two-observation problems (observer moves), set up two right triangles sharing a common height — solve simultaneously.
• Bearing problems: azimuth measured clockwise from North (0° to 360°); compass bearing uses N/S with E/W offset (e.g., N 35° E).
• Gradient or slope angle: tan θ = rise/run; used in road design and ramp problems.
• The angle of elevation from A to B equals the angle of depression from B to A (alternate interior angles).

Definitions

Common mistakes

• Confusing angle of elevation with angle of depression — draw a diagram first, always.
• Setting up d as the slant distance instead of the horizontal distance — h = d sin α, not h = d tan α, if d is the hypotenuse.
• In two-observation problems, using the wrong formula when the observer moves away instead of toward the object.
• Forgetting that bearings are from North (not East) when converting to standard mathematical angles.

Formulas

Exam tips

• Memorize the case-selection rule: SAS or SSS → Law of Cosines first; anything with two angles → Law of Sines.
• After finding two angles by Law of Sines, get the third by subtraction from 180° — never apply the Law of Sines a third time (accumulates rounding error).
• In board multiple-choice, if two answer choices look like an ambiguous case pair, the problem likely has two solutions — read carefully for which one is requested.
• When sides are given as integers, the answer often involves a clean radical (√52 = 2√13) — recognize these to select quickly from MCQ choices.
• Use Law of Cosines rearranged (cosC formula) to identify obtuse triangles: negative cosine means angle > 90°.

Key points

• Oblique triangles have NO right angle. Two tools: Law of Sines (angle-side pairs) and Law of Cosines (two sides + included angle, or all three sides).
• Case selection guide: SAA/ASA → Law of Sines; SAS/SSS → Law of Cosines; SSA → Law of Sines but check for ambiguous case.
• Law of Sines: a/sinA = b/sinB = c/sinC = 2R, where R is the circumradius.
• Law of Cosines: c² = a² + b² − 2ab cosC. Rearrange to find angle: cosC = (a² + b² − c²)/(2ab).
• Ambiguous case (SSA): given side a, side b, and angle A. Up to two triangles may exist. Always check if sin B > 1 (no solution) or if two values of B are valid.
• Sum of angles in any plane triangle = 180°. Use this to find the third angle after two are known.
• The largest angle is always opposite the longest side — use for checking solutions.

Definitions

Common mistakes

• Using Law of Sines for an SAS case — only Law of Cosines works when the given angle is between the two given sides.
• In the SSA case, checking only one value of B and missing the second valid triangle.
• Applying the Law of Cosines with the wrong angle: C must be the angle INCLUDED between sides a and b.
• Rounding intermediate values — always carry at least 4 significant figures through the calculation; round only the final answer.
• Forgetting that arcsin returns values in [−90°, 90°] — the obtuse solution B₂ = 180° − B₁ must be checked manually.

Formulas

Exam tips

• For a board item giving all three sides, default to Heron's formula — compute s first, then each bracket term.
• A 3-4-5 right triangle has area = 6, r = 1, R = 2.5 — memorize these benchmark values for speed.
• If a problem gives SAS, use A = ½ ab sinC first, then Law of Cosines to find c if needed — this sequence avoids extra steps.
• Area items often appear as the final part of a multi-step triangle problem — solve for the missing elements first, then apply the area formula.

Key points

• Three area formulas cover all triangle configurations: SAS form, Heron's formula (SSS), and base-height form.
• SAS area: A = ½ab sinC — the most commonly tested area formula on the board.
• Heron's formula: A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2 is the SEMI-perimeter.
• The area formula A = ½ base × height applies only when the height is explicitly given or easily computed.
• Inscribed circle radius (inradius): r = Area / s; appears in board items pairing area with the inscribed circle.
• Circumscribed circle radius: R = abc / (4A); connects all three sides with the circumcircle.
• For a triangle inscribed in a circle or circumscribing a circle, area problems often combine the area formula with R or r formulas.

Definitions

Common mistakes

• Using s = a + b + c (full perimeter) instead of s = (a+b+c)/2 in Heron's formula — this is the single most common Heron error.
• In A = ½ ab sinC, using a non-included angle — C must be BETWEEN sides a and b.
• Forgetting to take the square root at the end of Heron's formula.
• Confusing inradius r with circumradius R — remember r = A/s and R = abc/(4A).

Formulas

Exam tips

• Most board items on spherical trigonometry test only the spherical excess and area formula — master these two formulas first.
• For area problems: find E, plug into Area = πR²E/180° — that is all.
• If given a right spherical triangle (C = 90°), you can use cos c = cos a cos b directly (special case of Law of Cosines with cos C = 0).
• Angles in spherical triangles can exceed 90° — do not flag this as an error; it is geometrically valid.

Key points

• A spherical triangle is formed by three great-circle arcs on the surface of a sphere. Sides (a, b, c) and angles (A, B, C) are all measured in angular units (degrees or radians).
• Unlike plane triangles, the sum of angles of a spherical triangle is ALWAYS greater than 180° and less than 540°.
• Spherical excess E = (A + B + C) − 180°. This is always positive and is the key to finding spherical triangle area.
• Spherical Law of Sines: sin a/sin A = sin b/sin B = sin c/sin C.
• Spherical Law of Cosines (for sides): cos a = cos b cos c + sin b sin c cos A.
• Right spherical triangles (one angle = 90°) are solved using Napier's Rules — useful shortcut for geodetic problems.
• Spherical trigonometry is used in geodesy (curved earth), astronomy, and navigation — on the board it appears primarily as area-of-spherical-triangle problems.
• Area of spherical triangle: Area = πR²E/180° where E is in degrees, or Area = R²E when E is in radians.

Definitions

Common mistakes

• Treating spherical triangles like plane triangles — the angle sum is NOT 180°; it is 180° + E.
• Using degrees in the area formula but forgetting to convert E to radians when the formula uses E_rad — always check which form is being applied.
• Applying the plane Law of Cosines (c² = a² + b² − 2ab cosC) to spherical triangles — spherical Law of Cosines uses cos a = cos b cos c + sin b sin c cos A (all cosines, no squares).
• Confusing angular side lengths with linear arc lengths — convert: arc length = Rθ (in radians).