Study guide
2.2 Trigonometry — Identities, Triangles & Spherical Trig
PRC Registered Electrical Engineer REE
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1 Chapter 1: Overview, RA 7920 & Electrical Engineering Ethics
1.1 PRC Board of Electrical Engineering & REE Examination Overview 1.2 Republic Act No. 7920 (New Electrical Engineering Law) & License Classes 1.3 Code of Ethics & Professional Practice for Electrical Engineers
2 Chapter 2: Math — Algebra, Trig & Geometry
2.1 Algebraic Equations, Polynomials & Complex Numbers 2.2 Trigonometry — Identities, Triangles & Spherical Trig 2.3 Analytic & Solid Geometry
3 Chapter 3: Engineering Mathematics — Calculus & Differential Equations
3.1 Differential Calculus, Limits, Derivatives & Applications 3.2 Integral Calculus, Integration Techniques & Area/Volume Applications 3.3 Ordinary Differential Equations & Initial-Value Problems
4 Chapter 4: Engineering Mathematics — Advanced Math & Transform Methods
4.1 Laplace Transforms, Inverse Transforms & Circuit Applications 4.2 Fourier Series, Fourier Transforms & Frequency-Domain Analysis 4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
5 Chapter 5: Engineering Mathematics — Probability, Statistics & Engineering Data Analysis
5.1 Probability Theory, Counting Techniques & Bayes' Theorem 5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions 5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
6 Chapter 6: Engineering Mathematics — Numerical Methods & Analysis
6.1 Numerical Root-Finding & Solution of Linear Systems 6.2 Interpolation, Numerical Integration & Numerical Solution of ODEs
7 Chapter 7: ESAS — Physics, Chemistry & Computer Fundamentals
7.1 Physics for Engineers 7.2 General Chemistry & Materials Science 7.3 Computer Fundamentals, Programming & Data Logic 7.4 Logic Circuits, Switching Theory & Microprocessor Systems
8 Chapter 8: ESAS — Mechanics & Thermal-Fluid Engineering
8.1 Engineering Mechanics — Statics & Dynamics 8.2 Strength of Materials & Mechanics of Deformable Bodies 8.3 Thermal-Fluid Engineering — Thermodynamics & Fluid Mechanics
9 Chapter 9: ESAS — Engineering Economics, Management & Contracts
9.1 Engineering Economics & Depreciation 9.2 Engineering Management & Ethics 9.3 Philippine Electrical Engineering Law, Code & Contracts 9.4 Technopreneurship 101 & Management of Engineering Projects
10 Chapter 10: PEC 1 — Safety, Sizing & Protection
10.1 PEC Fundamentals, Safety Standards & Grounding/Bonding 10.2 Conductor Sizing, Ampacity Adjustment & Voltage Drop 10.3 Overcurrent Protection, Branch Circuit & Feeder Sizing 10.4 Basic Occupational Safety & Health (BOSH) and Electrical Safety Standards
11 Chapter 11: PEC 1 & 2 — Equipment & Utility Systems
11.1 PEC Equipment Installation & Wiring Methods 11.2 Utility Systems, Transformers & Switchgear 11.3 Special Occupancies & Hazardous Locations
12 Chapter 12: Electrical Professional — DC & AC Circuit Fundamentals
12.1 DC Circuit Analysis & Network Theorems 12.2 Single-Phase AC Circuits & Phasor Analysis 12.3 Balanced & Unbalanced Three-Phase AC Circuits
13 Chapter 13: Electrical Professional — EM Fields & Transients
13.1 Electrostatics, Magnetostatics & Maxwell's Equations 13.2 Electromagnetic Waves & Transmission Line Fields 13.3 First-Order & Second-Order Circuit Transients
14 Chapter 14: Electrical Professional — Electronics, Industrial Electronics & Communications
14.1 Electronics 1 & 2 — Semiconductor Devices & Analog Circuits 14.2 Electrical Apparatus & Devices — Industrial Electronics & Power Converters 14.3 Fundamentals of Electronic Communications
15 Chapter 15: Electrical Professional — Transformers & DC Machines
15.1 Single-Phase & Three-Phase Transformers 15.2 Transformer Connections & Special Transformers 15.3 DC Generators & DC Motors
16 Chapter 16: Electrical Professional — AC Synchronous & Induction Machinery
16.1 Synchronous Generators (Alternators) 16.2 Synchronous Motors & Special AC Machines 16.3 Three-Phase & Single-Phase Induction Motors
17 Chapter 17: Electrical Professional — Power Plants & Energy Conversion
17.1 Hydroelectric, Thermal & Combined Cycle Power Plants 17.2 Renewable Energy Systems & Geothermal Power Plants 17.3 Power Plant Economics, Load Curves & Energy Conversion
18 Chapter 18: Electrical Professional — Transmission & Distribution
18.1 Transmission Line Parameters & Performance Models 18.2 Mechanical Overhead Line Design, Underground Cables & Insulators 18.3 Distribution Systems, Substation Layouts & Power Flow Control
19 Chapter 19: Electrical Professional — Power System Analysis
19.1 The Per-Unit System, Base Conversion & Impedance Diagrams 19.2 Symmetrical Components & Unbalanced Fault Analysis 19.3 Load Flow, Economic Dispatch & Power System Stability
20 Chapter 20: Electrical Professional — Protection, Control, Instrumentation & Illumination
20.1 Power System Protection & Protective Relaying 20.2 Industrial Motor Control, Logic Circuits & PLC Automation 20.3 Electrical Instrumentation, Transducers & Measurement Systems 20.4 Illumination Engineering, Lighting Design & Photometrics 20.5 Feedback Control Systems & Research Methods
2.2 Trigonometry — Identities, Triangles & Spherical Trig
Key Takeaways Explain This Topic AI
Fundamental Pythagorean, sum/difference, double-angle, and half-angle trigonometric identities are essential for simplifying complex AC sinusoidal waveforms.
The Law of Sines ( a / sin A = b / sin B = c / sin C = 2 R a/\sin A = b/\sin B = c/\sin C = 2R a / sin A = b / sin B = c / sin C = 2 R ) and Law of Cosines ( c 2 = a 2 + b 2 − 2 a b cos C c^2 = a^2 + b^2 - 2ab \cos C c 2 = a 2 + b 2 − 2 ab cos C ) solve oblique plane triangles in surveying and vector dynamics.
Plane triangle area can be calculated via SAS formula ( 0.5 a b sin C 0.5 ab \sin C 0.5 ab sin C ), Heron's formula ( s ( s − a ) ( s − b ) ( s − c ) \sqrt{s(s-a)(s-b)(s-c)} s ( s − a ) ( s − b ) ( s − c ) ), or circumradius ( a b c / 4 R abc / 4R ab c /4 R ).
Spherical trigonometry governs spherical surface geometry where interior angles sum to between 180 ∘ 180^\circ 18 0 ∘ and 540 ∘ 540^\circ 54 0 ∘ , with area proportional to spherical excess E = A + B + C − 180 ∘ E = A + B + C - 180^\circ E = A + B + C − 18 0 ∘ .
Napier's Rules for right spherical triangles streamline calculations using the 'circle of five parts' and SIN-COOP / SIN-TAD mnemonics.
Last updated: August 2026
2.2 Trigonometry — Identities, Triangles & Spherical Trig
Trigonometry is indispensable in electrical engineering for solving plane geometric problems, surveying power line routes, and analyzing alternating current (AC) voltage and current waveforms. Spherical trigonometry extends these concepts to three-dimensional spherical surfaces, critical for terrestrial navigation, power grid satellite positioning, and global geographic calculations.
1. Fundamental Trigonometric Identities
Pythagorean Identities sin 2 θ + cos 2 θ = 1 \sin^2 \theta + \cos^2 \theta = 1 sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + \tan^2 \theta = \sec^2 \theta 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ 1 + \cot^2 \theta = \csc^2 \theta 1 + cot 2 θ = csc 2 θ
Angle Addition & Subtraction Formulas sin ( α ± β ) = sin α cos β ± cos α sin β \sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta sin ( α ± β ) = sin α cos β ± cos α sin β cos ( α ± β ) = cos α cos β ∓ sin α sin β \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta cos ( α ± β ) = cos α cos β ∓ sin α sin β tan ( α ± β ) = tan α ± tan β 1 ∓ tan α tan β \tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta} tan ( α ± β ) = 1 ∓ tan α tan β tan α ± tan β
Double-Angle & Half-Angle Formulas
Category Double-Angle Formulas Half-Angle Formulas
Sine sin ( 2 θ ) = 2 sin θ cos θ \sin(2\theta) = 2 \sin\theta \cos\theta sin ( 2 θ ) = 2 sin θ cos θ sin ( θ 2 ) = ± 1 − cos θ 2 \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}} sin ( 2 θ ) = ± 2 1 − c o s θ
Cosine cos ( 2 θ ) = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ \cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta cos ( 2 θ ) = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ cos ( θ 2 ) = ± 1 + cos θ 2 \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}} cos ( 2 θ ) = ± 2 1 + c o s θ
Tangent tan ( 2 θ ) = 2 tan θ 1 − tan 2 θ \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta} tan ( 2 θ ) = 1 − t a n 2 θ 2 t a n θ tan ( θ 2 ) = 1 − cos θ sin θ = sin θ 1 + cos θ \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta} tan ( 2 θ ) = s i n θ 1 − c o s θ = 1 + c o s θ s i n θ
Power Reduction Formulas in AC Circuits: sin 2 θ = 1 − cos ( 2 θ ) 2 , cos 2 θ = 1 + cos ( 2 θ ) 2 \sin^2 \theta = \frac{1 - \cos(2\theta)}{2}, \quad \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} sin 2 θ = 2 1 − c o s ( 2 θ ) , cos 2 θ = 2 1 + c o s ( 2 θ ) These identities allow electrical engineers to compute instantaneous power and average power in AC circuits.
2. Oblique Plane Triangle Solutions
For any plane triangle with sides a , b , c a, b, c a , b , c opposite to angles A , B , C A, B, C A , B , C :
Law of Sines a sin A = b sin B = c sin C = 2 R \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R sin A a = sin B b = sin C c = 2 R
where R R R is the radius of the circumscribed circle (circumradius).
Law of Cosines a 2 = b 2 + c 2 − 2 b c cos A a^2 = b^2 + c^2 - 2bc \cos A a 2 = b 2 + c 2 − 2 b c cos A b 2 = a 2 + c 2 − 2 a c cos B b^2 = a^2 + c^2 - 2ac \cos B b 2 = a 2 + c 2 − 2 a c cos B c 2 = a 2 + b 2 − 2 a b cos C c^2 = a^2 + b^2 - 2ab \cos C c 2 = a 2 + b 2 − 2 ab cos C
Triangle Area Formulas
Side-Angle-Side (SAS): Area = 1 2 a b sin C = 1 2 b c sin A = 1 2 a c sin B \text{Area} = \frac{1}{2} ab \sin C = \frac{1}{2} bc \sin A = \frac{1}{2} ac \sin B Area = 2 1 ab sin C = 2 1 b c sin A = 2 1 a c sin B
Heron's Formula (Three Sides - SSS): Area = s ( s − a ) ( s − b ) ( s − c ) \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} Area = s ( s − a ) ( s − b ) ( s − c ) where the semi-perimeter s = a + b + c 2 s = \frac{a + b + c}{2} s = 2 a + b + c .
In terms of Circumradius ( R R R ) and Inradius ( r r r ): Area = a b c 4 R = r ⋅ s \text{Area} = \frac{abc}{4R} = r \cdot s Area = 4 R ab c = r ⋅ s
3. Spherical Trigonometry Fundamentals
A spherical triangle is formed on the surface of a sphere by three intersecting arcs of great circles.
Key Properties of Spherical Triangles
Sides a , b , c a, b, c a , b , c are measured as central angles (in degrees or radians), not linear distances.
Interior angles A , B , C A, B, C A , B , C satisfy: 180 ∘ < A + B + C < 540 ∘ 180^\circ < A + B + C < 540^\circ 18 0 ∘ < A + B + C < 54 0 ∘
Spherical Excess ( E E E ): The amount by which the sum of interior angles exceeds 180 ∘ 180^\circ 18 0 ∘ : E = A + B + C − 180 ∘ E = A + B + C - 180^\circ E = A + B + C − 18 0 ∘
Area of a Spherical Triangle: Area = π R 2 E ∘ 180 ∘ = R 2 E rad \text{Area} = \frac{\pi R^2 E^\circ}{180^\circ} = R^2 E_{\text{rad}} Area = 18 0 ∘ π R 2 E ∘ = R 2 E rad where R R R is the sphere's radius and E ∘ E^\circ E ∘ is in degrees.
Fundamental Spherical Laws
Spherical Law of Sines: sin a sin A = sin b sin B = sin c sin C \frac{\sin a}{\sin A} = \frac{\sin b}{\sin B} = \frac{\sin c}{\sin C} 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 \cos a = \cos b \cos c + \sin b \sin c \cos A cos a = cos b cos c + sin b sin c cos A
Spherical Law of Cosines for Angles: cos A = − cos B cos C + sin B sin C cos a \cos A = -\cos B \cos C + \sin B \sin C \cos a cos A = − cos B cos C + sin B sin C cos a
4. Napier's Rules for Right Spherical Triangles
For a right spherical triangle where C = 90 ∘ C = 90^\circ C = 9 0 ∘ , John Napier devised a simple rule utilizing a "circle of five parts":
- The Five Parts (Arranged in Circle)
- Side a a a
- Side b b b
- Complement of Angle A A A (denoted co- A = 90 ∘ − A \text{co-}A = 90^\circ - A co- A = 9 0 ∘ − A )
- Complement of Hypotenuse c c c (denoted co- c = 90 ∘ − c \text{co-}c = 90^\circ - c co- c = 9 0 ∘ − c )
- Complement of Angle B B B (denoted co- B = 90 ∘ − B \text{co-}B = 90^\circ - B co- B = 9 0 ∘ − B )
- The Two Fundamental Rules
Rule 1 (SIN-COOP): The sine of any middle part is equal to the product of the cosines of the opposite parts. sin ( middle part ) = cos ( opposite 1 ) ⋅ cos ( opposite 2 ) \sin(\text{middle part}) = \cos(\text{opposite}_1) \cdot \cos(\text{opposite}_2) sin ( middle part ) = cos ( opposite 1 ) ⋅ cos ( opposite 2 )
Rule 2 (SIN-TAD): The sine of any middle part is equal to the product of the tangents of the adjacent parts. sin ( middle part ) = tan ( adjacent 1 ) ⋅ tan ( adjacent 2 ) \sin(\text{middle part}) = \tan(\text{adjacent}_1) \cdot \tan(\text{adjacent}_2) sin ( middle part ) = tan ( adjacent 1 ) ⋅ tan ( adjacent 2 )
- Solved Practice Examples
- Example 1: Oblique Triangle & Area via Heron's Formula
Problem: A transmission tower guy-wire anchor layout forms a triangle with side lengths a = 7 m a = 7\ \text{m} a = 7 m , b = 8 m b = 8\ \text{m} b = 8 m , and c = 9 m c = 9\ \text{m} c = 9 m . Find the area of the triangular plot and the largest interior angle.
Solution:
Compute semi-perimeter s s s : s = 7 + 8 + 9 2 = 12 m s = \frac{7 + 8 + 9}{2} = 12\ \text{m} s = 2 7 + 8 + 9 = 12 m
Calculate area using Heron's formula: Area = 12 ( 12 − 7 ) ( 12 − 8 ) ( 12 − 9 ) = 12 × 5 × 4 × 3 = 720 ≈ 26.8328 m 2 \text{Area} = \sqrt{12(12 - 7)(12 - 8)(12 - 9)} = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} \approx 26.8328\ \text{m}^2 Area = 12 ( 12 − 7 ) ( 12 − 8 ) ( 12 − 9 ) = 12 × 5 × 4 × 3 = 720 ≈ 26.8328 m 2
Find largest angle C C C (opposite longest side c = 9 m c = 9\ \text{m} c = 9 m ) using Law of Cosines: c 2 = a 2 + b 2 − 2 a b cos C ⟹ 9 2 = 7 2 + 8 2 − 2 ( 7 ) ( 8 ) cos C c^2 = a^2 + b^2 - 2ab \cos C \implies 9^2 = 7^2 + 8^2 - 2(7)(8) \cos C c 2 = a 2 + b 2 − 2 ab cos C ⟹ 9 2 = 7 2 + 8 2 − 2 ( 7 ) ( 8 ) cos C 81 = 49 + 64 − 112 cos C ⟹ 81 = 113 − 112 cos C 81 = 49 + 64 - 112 \cos C \implies 81 = 113 - 112 \cos C 81 = 49 + 64 − 112 cos C ⟹ 81 = 113 − 112 cos C 112 cos C = 32 ⟹ cos C = 32 112 = 2 7 ≈ 0.2857 112 \cos C = 32 \implies \cos C = \frac{32}{112} = \frac{2}{7} \approx 0.2857 112 cos C = 32 ⟹ cos C = 112 32 = 7 2 ≈ 0.2857 C = arccos ( 0.2857 ) ≈ 73.40 ∘ C = \arccos(0.2857) \approx 73.40^\circ C = arccos ( 0.2857 ) ≈ 73.4 0 ∘
Example 2: Napier's Rule Application
Problem: In a right spherical triangle with C = 90 ∘ C = 90^\circ C = 9 0 ∘ , given side a = 45 ∘ a = 45^\circ a = 4 5 ∘ and side b = 60 ∘ b = 60^\circ b = 6 0 ∘ , find the hypotenuse arc c c c .
Solution:
Select middle part co- c = 90 ∘ − c \text{co-}c = 90^\circ - c co- c = 9 0 ∘ − c . Its opposite parts on Napier's circle are sides a a a and b b b .
Apply SIN-COOP rule: sin ( co- c ) = cos ( a ) ⋅ cos ( b ) \sin(\text{co-}c) = \cos(a) \cdot \cos(b) sin ( co- c ) = cos ( a ) ⋅ cos ( b )
Recall sin ( 90 ∘ − c ) = cos c \sin(90^\circ - c) = \cos c sin ( 9 0 ∘ − c ) = cos c : cos c = cos ( 45 ∘ ) ⋅ cos ( 60 ∘ ) = ( 2 2 ) ⋅ ( 1 2 ) = 2 4 ≈ 0.35355 \cos c = \cos(45^\circ) \cdot \cos(60^\circ) = \left(\frac{\sqrt{2}}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{\sqrt{2}}{4} \approx 0.35355 cos c = cos ( 4 5 ∘ ) ⋅ cos ( 6 0 ∘ ) = ( 2 2 ) ⋅ ( 2 1 ) = 4 2 ≈ 0.35355
Solve for c c c : c = arccos ( 0.35355 ) ≈ 69.30 ∘ c = \arccos(0.35355) \approx 69.30^\circ c = arccos ( 0.35355 ) ≈ 69.3 0 ∘