Plane and Spherical Trigonometry - Cheat Sheet
Your last-minute revision companion for Plane and Spherical Trigonometry. Covers functions, identities, oblique triangles, areas, and spherical trigonometry — everything you need for the PRC Civil Engineer Licensure Examination.
Fundamental Trigonometric Functions & Identities
Formulas
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Meaning
sin, cos, tan = sine, cosine, tangent of angle θ; opp = opposite side, adj = adjacent side, hyp = hypotenuse (right triangle)
Watch out
Remember the mnemonic SOH-CAH-TOA; watch for angle units (degrees vs radians on calculator)
When to use
Whenever you need to relate an angle to the sides of a right triangle
sin²θ + cos²θ = 1
Meaning
Pythagorean identity: the fundamental trigonometric identity
Watch out
This is ALWAYS true; never assume sin²θ + cos²θ = anything other than 1
When to use
To eliminate sin or cos from an equation, or to verify trigonometric expressions
tan θ = sin θ / cos θ
Meaning
tan θ is the ratio of sine to cosine
Watch out
tan θ is undefined when cos θ = 0 (θ = 90°, 270°, etc.)
When to use
When converting between tangent and sine/cosine forms; when cos θ ≠ 0
1 + tan²θ = sec²θ
Meaning
sec²θ = 1 + tan²θ; sec θ = 1/cos θ (secant)
Watch out
This is derived from sin²θ + cos²θ = 1 divided by cos²θ
When to use
To eliminate tangent or express in secant form
1 + cot²θ = csc²θ
Meaning
csc²θ = 1 + cot²θ; csc θ = 1/sin θ (cosecant), cot θ = 1/tan θ (cotangent)
Watch out
Rarely tested in board exams but included for completeness
When to use
When working with cotangent or cosecant forms
sin 2θ = 2 sin θ cos θ
Meaning
Double-angle formula for sine
Watch out
This is NOT sin²θ; do not confuse with the Pythagorean identity
When to use
When an angle is twice another angle, or when simplifying expressions with 2θ
cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
Meaning
Three equivalent forms of the double-angle formula for cosine
Watch out
All three forms are correct; pick the one that avoids extra work
When to use
Choose the form that matches what you know: use cos²θ − sin²θ if both known, use 1 − 2sin²θ if only sin known, use 2cos²θ − 1 if only cos known
tan 2θ = 2 tan θ / (1 − tan²θ)
Meaning
Double-angle formula for tangent
Watch out
Denominator is 1 − tan²θ (NOT 1 + tan²θ); undefined when tan²θ = 1
When to use
When dealing with tangent of a double angle
sin(A ± B) = sin A cos B ± cos A sin B
Meaning
Angle addition/subtraction formula for sine
Watch out
Sign between terms matches the sign between A and B; the ± on right side IS the same as the ± on left
When to use
When you have sin of sum or difference of two angles
cos(A ± B) = cos A cos B ∓ sin A sin B
Meaning
Angle addition/subtraction formula for cosine
Watch out
Sign between terms is OPPOSITE to the sign between A and B (∓); cos sum uses minus, cos difference uses plus internally
When to use
When you have cos of sum or difference of two angles
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
Meaning
Angle addition/subtraction formula for tangent
Watch out
Denominator sign is OPPOSITE to numerator sign; watch for division by zero
When to use
When dealing with tangent of sum or difference of two angles
Common values
Value
0.5 or 1/2
Symbol
sin(π/6)
Quantity
sin 30°
Value
√2/2 ≈ 0.7071
Symbol
sin(π/4)
Quantity
sin 45°
Value
√3/2 ≈ 0.8660
Symbol
sin(π/3)
Quantity
sin 60°
Value
√3/2 ≈ 0.8660
Symbol
cos(π/6)
Quantity
cos 30°
Value
√2/2 ≈ 0.7071
Symbol
cos(π/4)
Quantity
cos 45°
Value
0.5 or 1/2
Symbol
cos(π/3)
Quantity
cos 60°
Value
1/√3 ≈ 0.5774
Symbol
tan(π/6)
Quantity
tan 30°
Value
1
Symbol
tan(π/4)
Quantity
tan 45°
Value
√3 ≈ 1.732
Symbol
tan(π/3)
Quantity
tan 60°
Value
1.414
Symbol
≈1.414
Quantity
√2
Value
1.732
Symbol
≈1.732
Quantity
√3
Value
3.14159...
Symbol
≈3.14159
Quantity
π
Important facts
- sin, cos, tan are periodic: sin and cos have period 360° (2π rad); tan has period 180° (π rad)
- ASTC rule: All positive (Q1), Sin positive (Q2), Tan positive (Q3), Cos positive (Q4)
- sin 0° = 0, sin 30° = 0.5, sin 45° = √2/2 ≈ 0.7071, sin 60° = √3/2 ≈ 0.8660, sin 90° = 1
- cos 0° = 1, cos 30° = √3/2 ≈ 0.8660, cos 45° = √2/2 ≈ 0.7071, cos 60° = 0.5, cos 90° = 0
- tan 0° = 0, tan 30° = 1/√3 ≈ 0.5774, tan 45° = 1, tan 60° = √3 ≈ 1.732, tan 90° = undefined
- Complementary angles: sin θ = cos(90° − θ); tan θ = cot(90° − θ)
- Supplementary angles: sin θ = sin(180° − θ); cos θ = −cos(180° − θ)
Key definitions
Radian
Example
π/4 radians = 45°, π/2 radians = 90°
Unit of angle measure where 1 radian = 180°/π ≈ 57.3°; full circle = 2π radians
Reference Angle
Example
Reference angle for 150° is 30°; sin 150° = sin 30° = 0.5
Acute angle between the terminal side of an angle and the x-axis; used to find trig values in all quadrants
Angle of Elevation
Example
Looking up at a tower top from ground level involves angle of elevation
Angle above the horizontal when looking upward from an observer to an object
Angle of Depression
Example
Looking down from a cliff to a boat involves angle of depression
Angle below the horizontal when looking downward from an observer to an object
Diagrams to know
- Right triangle with labeled opposite, adjacent, hypotenuse sides and angle θ
- Unit circle showing all four quadrants with angle positions
- Graph of sin θ, cos θ, tan θ over one complete cycle
Oblique Triangles — Law of Sines & Law of Cosines
Formulas
a/sin A = b/sin B = c/sin C
Meaning
Law of Sines: a, b, c = sides; A, B, C = angles opposite those sides respectively
Watch out
SSA case (two sides + angle not included) can produce 0, 1, or 2 valid triangles — check discriminant: b sin A vs a
When to use
When you know (angle + opposite side + another angle or side) OR two angles and any side (AAS, ASA, SSA cases)
c² = a² + b² − 2ab cos C
Meaning
Law of Cosines: c = side opposite angle C; a, b = other two sides; C = included angle between a and b
Watch out
Angle C MUST be included between sides a and b; if you have angle at wrong position, rearrange labeling; sign is MINUS not plus
When to use
When you know (two sides + included angle SAS) OR all three sides (SSS) — use to find unknown side or angle
a² = b² + c² − 2bc cos A
Meaning
Law of Cosines rearranged to find side a when angle A and sides b, c are known
Watch out
This is the same law as c² = a² + b² − 2ab cos C, just relabeled
When to use
Finding a side when you know two other sides and the angle opposite the unknown side
cos A = (b² + c² − a²) / (2bc)
Meaning
Law of Cosines rearranged to find angle A when all three sides a, b, c are known
Watch out
Make sure numerator is (other two sides squared minus opposite side squared); use arccos to get the angle
When to use
When SSS (all three sides given) — solve for each angle using this form
cos B = (a² + c² − b²) / (2ac)
Meaning
Law of Cosines to find angle B given all three sides
Watch out
Same structure as cos A formula; just relabel sides accordingly
When to use
Finding angle B in SSS case
cos C = (a² + b² − c²) / (2ab)
Meaning
Law of Cosines to find angle C given all three sides
Watch out
Once you have two angles, third angle = 180° − A − B (no need to recalculate)
When to use
Finding angle C in SSS case
Important facts
- Sum of angles in any triangle = 180° (or π radians)
- In Law of Sines, the ratio a/sin A is constant for all three side-angle pairs in one triangle
- Law of Cosines reduces to Pythagorean theorem when C = 90°: c² = a² + b² (since cos 90° = 0)
- SSA (ambiguous case) critical values: h = b sin A (minimum valid a); if a < h no solution; if a = h one right triangle; if h < a < b two solutions; if a ≥ b one solution
- To solve a triangle completely, find all three sides and all three angles
- Always check: sum of angles = 180°; largest side opposite largest angle; smallest side opposite smallest angle
Key definitions
Oblique Triangle
Example
Triangle with sides 5, 7, 8 is oblique because no angle is 90°
Any triangle that does NOT have a 90° angle (neither right-angled); solved using Law of Sines or Law of Cosines
Included Angle
Example
In triangle ABC, if you know sides a and b, angle C is included; if you know angle C, sides a and b must be the ones bounding it
The angle between two known sides; the angle that 'sits between' the two sides you're using
Ambiguous Case (SSA)
Example
Given a, b, and angle A: if a < b sin A then no triangle; if a = b sin A then one right triangle; if a > b sin A then two triangles
When two sides and an angle (not included) are known; can yield 0, 1, or 2 solutions depending on whether b sin A < a, b sin A = a, or a < b sin A
Triangle Solution Cases
Example
SAS case uses Law of Cosines; AAS case uses Law of Sines
ASA (two angles + included side), AAS (two angles + non-included side), SAS (two sides + included angle), SSS (three sides), SSA (two sides + non-included angle)
Diagrams to know
- Oblique triangle with labeled sides a, b, c and angles A, B, C showing opposite relationships
- Ambiguous case diagram showing two possible triangles for given a, b, and angle A
- Decision tree: when to use Law of Sines vs Law of Cosines based on given information
Triangle Area Formulas
Formulas
Area = (1/2) ab sin C
Meaning
Area using two sides (a, b) and included angle C between them
Watch out
Angle C MUST be between sides a and b; if angle is not included, you cannot use this formula directly
When to use
When you know two sides and the included angle (SAS case); fastest method for area when angle is known
Area = (1/2) ac sin B
Meaning
Area using sides a, c and included angle B
Watch out
All three forms are equivalent; choose based on which two sides and angle you know
When to use
Same as above but with different pair of sides and their included angle
Area = (1/2) bc sin A
Meaning
Area using sides b, c and included angle A
Watch out
Remember: two sides × sin(included angle) × (1/2)
When to use
Same principle as the two formulas above
Area = √[s(s−a)(s−b)(s−c)]
Meaning
Heron's Formula: s = (a+b+c)/2 is the semi-perimeter; a, b, c are the three sides
Watch out
CRITICAL: s is the SEMI-perimeter (half the perimeter), NOT the full perimeter; (s−a), (s−b), (s−c) must ALL be positive or formula fails
When to use
When all three sides (SSS) are known and you need area; do NOT use if angles are known
s = (a+b+c)/2
Meaning
Semi-perimeter: half the sum of all three sides
Watch out
This is semi-perimeter, not full perimeter; forgetting this is the #1 Heron's formula error
When to use
Always compute s first before using Heron's formula
Area = (1/2) × base × height
Meaning
Basic area formula using perpendicular height from base
Watch out
Height must be perpendicular to the base; if not given, may need to compute using sin or cos
When to use
When height is known or can be computed; used in applications like surveying
Common values
Value
√3/4 ≈ 0.433
Symbol
for unit side
Quantity
Equilateral triangle area factor
Important facts
- For a right triangle with legs a and b: Area = (1/2) ab (special case since sin 90° = 1)
- Heron's formula works for ANY triangle as long as you know all three sides
- Equilateral triangle with side a: Area = (√3/4) a²
- If one angle is obtuse (>90°), use sin C formula since sin(obtuse angle) is still positive
- Check area calculation: Area should be positive and less than (1/2) × product of any two sides
Key definitions
Semi-perimeter
Example
Triangle with sides 3, 4, 5 has s = (3+4+5)/2 = 6
Half of the triangle's perimeter; s = (a+b+c)/2; essential for Heron's formula
Heron's Formula
Example
Triangle with sides 3, 4, 5: s = 6; Area = √[6(6−3)(6−4)(6−5)] = √[6×3×2×1] = √36 = 6
Method to compute triangle area from all three sides without needing angles or height
Included Angle
Example
For Area = (1/2) ab sin C, angle C is the included angle between sides a and b
The angle between two sides used in the (1/2)ab sin C area formula
Diagrams to know
- Triangle with height drawn perpendicular to base, showing Area = (1/2) × base × h
- Triangle with two sides and included angle marked, showing (1/2)ab sin C concept
Spherical Trigonometry (Essentials)
Formulas
sin a / sin A = sin b / sin B = sin c / sin C
Meaning
Spherical Law of Sines: a, b, c are sides (as angular measures in degrees/radians); A, B, C are angles at vertices
Watch out
Sides are measured as angles subtended at sphere center, not as linear distances; applies ONLY to spherical triangles
When to use
Spherical triangles (on a sphere surface) analogous to plane law of sines; used in geodesy and navigation
cos a = cos b cos c + sin b sin c cos A
Meaning
Spherical Law of Cosines: a is the side opposite angle A; b, c are adjacent sides; A is the angle between b and c
Watch out
Structure is similar to plane law of cosines but uses all sines and cosines; not simple subtraction as in plane case
When to use
Finding unknown side a when two sides b, c and included angle A are known on a sphere
cos A = −cos B cos C + sin B sin C cos a
Meaning
Spherical Law of Cosines for angles: A is the angle opposite side a; B, C are adjacent angles
Watch out
NOTE the MINUS sign in front (unlike plane law); critical difference from the side formula
When to use
Finding an angle when all three sides are known (SSS case) on a sphere
Spherical Excess E = A + B + C − 180°
Meaning
E is the excess of the sum of angles over 180° (in degrees); in radians: E = A + B + C − π
Watch out
E is ALWAYS positive for spherical triangles; E = 0 only in the limit as sphere radius → ∞ (becomes plane triangle)
When to use
To find the area of a spherical triangle when all angles are known
Area of Spherical Triangle = (πR² E) / 180° = R² E (in radians)
Meaning
Area on sphere of radius R with spherical excess E; first form if E in degrees, second if E in radians
Watch out
E MUST be converted to radians if using second formula; R² units are same as final area units (e.g., m²)
When to use
When you have the three angles of a spherical triangle and sphere radius; use to compute surface area
Important facts
- Sum of angles in a spherical triangle is ALWAYS > 180° (unlike plane triangles which = 180°)
- Spherical triangles appear in navigation (great circle routes), geodesy (surveying large areas), and astronomy
- For small spheres (R → ∞ relative to triangle size), spherical formulas approach plane triangle formulas
- Each side of a spherical triangle is less than 180° (or π radians); if equal to 180° it becomes a full great circle
- Correspondence: Spherical angle A ↔ Plane angle A; Spherical side a ↔ Plane opposite side a
Key definitions
Spherical Triangle
Example
Geodetic triangle formed by three cities on Earth's surface (treated as sphere)
A triangle on the surface of a sphere whose sides are arcs of great circles; angles and sides are measured as angles subtended at sphere center
Great Circle
Example
Earth's equator is a great circle; latitude lines (except equator) are small circles
A circle on a sphere whose center is at the sphere's center (e.g., equator, meridian); shortest path between two points on sphere
Spherical Excess
Example
Spherical triangle with angles 100°, 95°, 110° has E = 100° + 95° + 110° − 180° = 25°
The amount by which the sum of angles in a spherical triangle exceeds 180°; E = (A + B + C) − 180°
Angular Measure (on Sphere)
Example
An arc of 60° on a sphere of radius 6 m has actual length = (π/3) × 6 ≈ 6.28 m
Sides of a spherical triangle are measured as angles, not distances; e.g., a side of '30°' means the arc subtends 30° at the sphere's center
Diagrams to know
- Sphere with great circles and a spherical triangle marked with sides a, b, c and angles A, B, C
- Diagram showing spherical excess: how angles sum to more than 180° on a sphere
Common Problem-Solving Procedures
Formulas
Right Triangle: tan θ = opposite/adjacent; h = d tan θ
Meaning
h = height/distance along line of sight; d = horizontal distance; θ = angle of elevation or depression
Watch out
Angle θ is measured FROM horizontal; make sure you're using the correct angle (elevation vs depression); d must be horizontal distance
When to use
Surveying, heights of buildings/towers, angle-of-elevation/depression problems
Verify triangle solution: A + B + C = 180°; largest side opposite largest angle
Meaning
Always check computed triangle angles sum to exactly 180°; side-angle ordering must be consistent
Watch out
Rounding errors can cause sum to be off by 0.1°–0.5°; this is acceptable; if > 1° recheck calculations
When to use
After solving any triangle to catch computational errors before finalizing answer
Ambiguous Case (SSA) Test: If a < b sin A, no triangle; if a = b sin A, one right triangle; if b sin A < a < b, two triangles; if a ≥ b, one triangle
Meaning
Given sides a, b and angle A (opposite side a), determine how many valid triangles exist
Watch out
Most common board-exam trap; always check this FIRST when given SSA data
When to use
Before solving SSA case, quickly check number of solutions to avoid missing a second solution or reporting impossible solution
Important facts
- When solving triangles, always work toward the unknowns in logical order: use given data to find one unknown, then use that to find next, etc.
- In surveying, bearings and azimuths are often given; convert to triangle angles using geometry (bearing angles often involve 90° adjustments)
- Angle of elevation from observer at point P to object at point Q is the angle above horizontal; angle of depression is below horizontal
- In engineering applications (NSCP 2015, AISC 360), trigonometry appears in member orientation angles, force resolution, and moment-arm calculations
Diagrams to know
- Angle of elevation and angle of depression diagrams
- Decision flowchart: which law (sines vs cosines) for each triangle case
Must remember
- 1. FUNDAMENTAL IDENTITY: sin²θ + cos²θ = 1 — ALWAYS TRUE, use to eliminate or verify trig expressions
- 2. LAW OF SINES: a/sin A = b/sin B = c/sin C — use when you have angle-opposite-side pairs (AAS, ASA, SSA); SSA is ambiguous — check first
- 3. LAW OF COSINES: c² = a² + b² − 2ab cos C — use for SAS (two sides + included angle) or SSS (all three sides); angle MUST be included between the sides
- 4. ANGLE SUM: A + B + C = 180° in plane triangles ONLY; verify after solving to catch errors
- 5. AREA FORMULAS: (1/2)ab sin C when two sides + included angle known; Heron's √[s(s−a)(s−b)(s−c)] when all three sides known — s = (a+b+c)/2 is SEMI-perimeter
- 6. SPECIAL ANGLES: sin 30° = 0.5, sin 45° = √2/2 ≈ 0.707, sin 60° = √3/2 ≈ 0.866; cos values reverse order: cos 30° = √3/2, cos 45° = √2/2, cos 60° = 0.5; tan 45° = 1
- 7. AMBIGUOUS CASE (SSA): Check b sin A vs a to determine number of solutions before solving; critical board-exam trap
- 8. SPHERICAL TRIANGLES: Angles sum to > 180°; area = πR²E/180° where E = (A+B+C)−180° is the spherical excess; sides measured as angles (not distances)
- 9. CALCULATOR UNITS: Always verify calculator is in DEGREES or RADIANS as needed; wrong unit = completely wrong answer
- 10. SIGN CONVENTIONS: In quadrants 2, 3, 4 some functions are negative (use ASTC rule: All, Sine, Tangent, Cosine); double-angle and sum formulas have specific ± patterns — memorize exact signs
Last minute tips
- TIP 1 — SSA AMBIGUITY CHECK: Given two sides a, b and angle A (opposite a), compute b sin A. If a < b sin A → no triangle; if a = b sin A → right triangle (one solution); if b sin A < a < b → TWO solutions exist (compute both angles B₁ and B₂, then C₁ and C₂); if a ≥ b → one solution. This catches the most common board-exam trap.
- TIP 2 — LAW OF COSINES SETUP: When given SAS, ALWAYS make sure the angle you're using is between the two sides. If the angle is NOT included, rearrange or use Law of Sines first. Forgetting this turns an easy problem into an unsolvable one.
- TIP 3 — HERON'S FORMULA TRAP: s = (a+b+c)/2 is the SEMI-perimeter. The most common error is using full perimeter. Also verify (s−a), (s−b), (s−c) are all positive; if any is zero or negative, the triangle doesn't exist.
- TIP 4 — ANGLE VERIFICATION: After solving any triangle, add up all three angles. They MUST sum to 180° (within 0.5° due to rounding). If they don't, you made a computational error — fix it before submitting. This takes 5 seconds and saves you from losing marks.
- TIP 5 — SPHERICAL VS PLANE: If the problem mentions 'Earth', 'sphere', 'great circle', or 'geodetic' → use spherical formulas. If it says 'flat', 'local', or 'surveying site' → use plane formulas. The spherical excess E = (A+B+C)−180° is your identifier.
Comparison tables
When to Use Law of Sines vs Law of Cosines
| Given Data (Case) | Use This Law | Formula to Apply | Notes |
|---|---|---|---|
| Two angles + any side (AAS, ASA) | Law of Sines | a/sin A = b/sin B = c/sin C | Find third angle (sum to 180°), then use sines |
| Two sides + included angle (SAS) | Law of Cosines | c² = a² + b² − 2ab cos C | Find third side, then use Law of Sines or Cosines for angles |
| Three sides (SSS) | Law of Cosines | cos A = (b² + c² − a²)/(2bc) | Find all three angles using this form; then verify sum = 180° |
| Two sides + non-included angle (SSA) | Law of Sines (with caution) | a/sin A = b/sin B | Check ambiguous case first; may have 0, 1, or 2 solutions |
Trigonometric Functions in All Four Quadrants
| Quadrant | Angle Range | sin θ | cos θ | tan θ | Mnemonic (Positive) |
|---|---|---|---|---|---|
| Q1 (First) | 0° to 90° | Positive | Positive | Positive | All |
| Q2 (Second) | 90° to 180° | Positive | Negative | Negative | Sine |
| Q3 (Third) | 180° to 270° | Negative | Negative | Positive | Tangent |
| Q4 (Fourth) | 270° to 360° | Negative | Positive | Negative | Cosine |
Special Angle Values (Common in Board Exams)
| Angle (degrees) | Angle (radians) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 0.5 | √3/2 ≈ 0.866 | 1/√3 ≈ 0.577 |
| 45° | π/4 | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 | 1 |
| 60° | π/3 | √3/2 ≈ 0.866 | 0.5 | √3 ≈ 1.732 |
| 90° | π/2 | 1 | 0 | Undefined |
| 180° | π | 0 | −1 | 0 |
Plane Triangles vs Spherical Triangles
| Property | Plane Triangle | Spherical Triangle | Key Difference |
|---|---|---|---|
| Sum of angles | A + B + C = 180° | A + B + C > 180° | Spherical always exceeds by E (excess) |
| Law of Sines | a/sin A = b/sin B = c/sin C | sin a/sin A = sin b/sin B = sin c/sin C | Spherical uses sin of sides (measured as angles) |
| Law of Cosines (sides) | c² = a² + b² − 2ab cos C | cos a = cos b cos c + sin b sin c cos A | Different structure for spherical formula |
| Area formula | (1/2)ab sin C or Heron's √[s(s−a)(s−b)(s−c)] | πR²E/180° or R²E (radians) | Spherical area depends on excess E and radius R |
| Sides measured as | Linear distances | Angular measures (degrees/radians) | Critical conceptual difference |
| Application | Local surveying, building layout, structural geometry | Geodesy, navigation, astronomy, large-scale surveying | Use spherical for Earth-scale problems (NSCP 2015 site surveys) |
Diagrams
Trigonometry Formula Family Tree
Overview of all major trigonometric formulas organized by category: fundamental identities, double angles, sum/difference, and triangle solution methods for both plane and spherical triangles.
Decision Tree: Which Triangle Formula to Use
Step-by-step decision flowchart to determine whether to use Law of Sines or Law of Cosines based on given triangle data (angles and sides). Includes SSA ambiguity check and case identification.
Solving an Oblique Triangle: State Progression
State diagram showing the progression of steps to solve an oblique triangle, from case identification through formula application to final verification that angles sum to 180°.