GELE Mathematics — Plane and Spherical TrigonometryCheat Sheet

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

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

Term

Radian

Example

π/4 radians = 45°, π/2 radians = 90°

Definition

Unit of angle measure where 1 radian = 180°/π ≈ 57.3°; full circle = 2π radians

Term

Reference Angle

Example

Reference angle for 150° is 30°; sin 150° = sin 30° = 0.5

Definition

Acute angle between the terminal side of an angle and the x-axis; used to find trig values in all quadrants

Term

Angle of Elevation

Example

Looking up at a tower top from ground level involves angle of elevation

Definition

Angle above the horizontal when looking upward from an observer to an object

Term

Angle of Depression

Example

Looking down from a cliff to a boat involves angle of depression

Definition

Angle below the horizontal when looking downward from an observer to an object

Formula

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)

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

Term

Oblique Triangle

Example

Triangle with sides 5, 7, 8 is oblique because no angle is 90°

Definition

Any triangle that does NOT have a 90° angle (neither right-angled); solved using Law of Sines or Law of Cosines

Term

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

Definition

The angle between two known sides; the angle that 'sits between' the two sides you're using

Term

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

Definition

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

Term

Triangle Solution Cases

Example

SAS case uses Law of Cosines; AAS case uses Law of Sines

Definition

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)

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

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

Value

√3/4 ≈ 0.433

Symbol

for unit side

Quantity

Equilateral triangle area factor

Term

Semi-perimeter

Example

Triangle with sides 3, 4, 5 has s = (3+4+5)/2 = 6

Definition

Half of the triangle's perimeter; s = (a+b+c)/2; essential for Heron's formula

Term

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

Definition

Method to compute triangle area from all three sides without needing angles or height

Term

Included Angle

Example

For Area = (1/2) ab sin C, angle C is the included angle between sides a and b

Definition

The angle between two sides used in the (1/2)ab sin C area formula

Formula

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

Formula

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

Formula

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

Formula

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

Formula

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

Term

Spherical Triangle

Example

Geodetic triangle formed by three cities on Earth's surface (treated as sphere)

Definition

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

Term

Great Circle

Example

Earth's equator is a great circle; latitude lines (except equator) are small circles

Definition

A circle on a sphere whose center is at the sphere's center (e.g., equator, meridian); shortest path between two points on sphere

Term

Spherical Excess

Example

Spherical triangle with angles 100°, 95°, 110° has E = 100° + 95° + 110° − 180° = 25°

Definition

The amount by which the sum of angles in a spherical triangle exceeds 180°; E = (A + B + C) − 180°

Term

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

Definition

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

Formula

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

Formula

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

Formula

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

Values

• Law of Sines
• a/sin A = b/sin B = c/sin C
• Find third angle (sum to 180°), then use sines

Property

Two angles + any side (AAS, ASA)

Values

• Law of Cosines
• c² = a² + b² − 2ab cos C
• Find third side, then use Law of Sines or Cosines for angles

Property

Two sides + included angle (SAS)

Values

• Law of Cosines
• cos A = (b² + c² − a²)/(2bc)
• Find all three angles using this form; then verify sum = 180°

Property

Three sides (SSS)

Values

• Law of Sines (with caution)
• a/sin A = b/sin B
• Check ambiguous case first; may have 0, 1, or 2 solutions

Property

Two sides + non-included angle (SSA)

Values

• 0° to 90°
• Positive
• Positive
• Positive
• All

Property

Q1 (First)

Values

• 90° to 180°
• Positive
• Negative
• Negative
• Sine

Property

Q2 (Second)

Values

• 180° to 270°
• Negative
• Negative
• Positive
• Tangent

Property

Q3 (Third)

Values

• 270° to 360°
• Negative
• Positive
• Negative
• Cosine

Property

Q4 (Fourth)

Values

• 0
• 0
• 1
• 0

Property

0°

Values

• π/6
• 0.5
• √3/2 ≈ 0.866
• 1/√3 ≈ 0.577

Property

30°

Values

• π/4
• √2/2 ≈ 0.707
• √2/2 ≈ 0.707
• 1

Property

45°

Values

• π/3
• √3/2 ≈ 0.866
• 0.5
• √3 ≈ 1.732

Property

60°

Values

• π/2
• 1
• 0
• Undefined

Property

90°

Values

• π
• 0
• −1
• 0

Property

180°

Values

• A + B + C = 180°
• A + B + C > 180°
• Spherical always exceeds by E (excess)

Property

Sum of angles

Values

• 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)

Property

Law of Sines

Values

• c² = a² + b² − 2ab cos C
• cos a = cos b cos c + sin b sin c cos A
• Different structure for spherical formula

Property

Law of Cosines (sides)

Values

• (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

Property

Area formula

Values

• Linear distances
• Angular measures (degrees/radians)
• Critical conceptual difference

Property

Sides measured as

Values

• Local surveying, building layout, structural geometry
• Geodesy, navigation, astronomy, large-scale surveying
• Use spherical for Earth-scale problems (NSCP 2015 site surveys)

Property

Application