Trigonometry - Revision notes
Trigonometry is the study of triangles and the relationships between their sides and angles. This chapter covers fundamental trigonometric concepts essential for UPCAT and other college entrance exams, including right triangle trigonometry, unit circle concepts, trigonometric identities, and the laws of sines and cosines. Master these concepts through step-by-step problem solving and practical applications.
Angle Measurement and Unit Circle
Formulas
Degree to Radian: θ_rad = (π × θ_deg)/180°
Example
Convert 45° to radians: θ_rad = (π × 45°)/180° = π/4 radians
Variables
θ_rad = angle in radians, θ_deg = angle in degrees
Application
Converting between angle measurements
Radian to Degree: θ_deg = (180° × θ_rad)/π
Example
Convert π/3 to degrees: θ_deg = (180° × π/3)/π = 60°
Variables
θ_deg = angle in degrees, θ_rad = angle in radians
Application
Converting radians back to degrees
Arc Length: s = rθ (θ in radians) or s = (πrθ)/180° (θ in degrees)
Example
Find arc length with r = 6 and θ = π/3: s = 6 × π/3 = 2π units
Variables
s = arc length, r = radius, θ = central angle
Application
Finding arc length on circles
Exam tips
- Memorize key conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2
- Draw unit circle diagrams to visualize angle positions
- Practice finding coterminal angles by adding/subtracting 360°
Key points
- Angles can be measured in degrees (360° = full rotation) or radians (2π = full rotation)
- Standard position: vertex at origin, initial side on positive x-axis
- Coterminal angles differ by multiples of 360° or 2π
- Reference angles are acute angles (0° to 90°) used to find trigonometric values
- Unit circle has radius = 1 and equation x² + y² = 1
Definitions
Standard Position
An angle positioned with vertex at origin and initial side along positive x-axis
Importance
Essential for defining trigonometric functions using coordinates
Reference Angle
The acute angle between the terminal side and the x-axis
Importance
Used to find trigonometric values in any quadrant
Common mistakes
- Forgetting to convert between degrees and radians
- Confusing coterminal angles with complementary angles
- Not identifying the correct reference angle for angles > 90°
Right Triangle Trigonometry
Formulas
sin θ = opposite/hypotenuse
Example
In triangle with opposite = 3, hypotenuse = 5: sin θ = 3/5 = 0.6
Variables
θ = angle, opposite = side opposite to angle, hypotenuse = longest side
Application
Finding ratios in right triangles
cos θ = adjacent/hypotenuse
Example
In triangle with adjacent = 4, hypotenuse = 5: cos θ = 4/5 = 0.8
Variables
θ = angle, adjacent = side next to angle, hypotenuse = longest side
Application
Finding ratios in right triangles
tan θ = opposite/adjacent
Example
In triangle with opposite = 3, adjacent = 4: tan θ = 3/4 = 0.75
Variables
θ = angle, opposite = side opposite to angle, adjacent = side next to angle
Application
Finding ratios without hypotenuse
Exam tips
- Draw and label triangles clearly before solving
- Memorize exact values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3
- Use SOH-CAH-TOA mnemonic to remember basic ratios
Key points
- SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
- Reciprocal functions: csc = 1/sin, sec = 1/cos, cot = 1/tan
- Special triangles: 30-60-90 and 45-45-90 triangles have specific ratios
- Pythagorean theorem: a² + b² = c² for right triangles
- Applications in real-world problems: heights, distances, angles of elevation/depression
Definitions
30-60-90 Triangle
Special right triangle with sides in ratio 1 : √3 : 2
Importance
Provides exact values for trigonometric functions at 30° and 60°
45-45-90 Triangle
Special right triangle with sides in ratio 1 : 1 : √2
Importance
Provides exact values for trigonometric functions at 45°
Common mistakes
- Confusing opposite and adjacent sides relative to the angle
- Using wrong triangle ratios for special angles
- Forgetting to rationalize denominators in exact answers
Trigonometric Functions and Identities
Formulas
sin²θ + cos²θ = 1
Example
If sin θ = 3/5, then cos²θ = 1 - (3/5)² = 1 - 9/25 = 16/25, so cos θ = ±4/5
Variables
θ = any angle
Application
Pythagorean identity - fundamental relationship
tan²θ + 1 = sec²θ
Example
If tan θ = 2, then sec²θ = 2² + 1 = 5, so sec θ = ±√5
Variables
θ = any angle where cos θ ≠ 0
Application
Pythagorean identity involving tangent and secant
cot²θ + 1 = csc²θ
Example
If cot θ = 1/3, then csc²θ = (1/3)² + 1 = 1/9 + 1 = 10/9, so csc θ = ±√10/3
Variables
θ = any angle where sin θ ≠ 0
Application
Pythagorean identity involving cotangent and cosecant
Exam tips
- Remember ASTC (All Students Take Calculus) for quadrant signs
- Use identities to solve for unknown trigonometric values
- Practice converting between different forms using identities
Key points
- Six trigonometric functions: sin, cos, tan, csc, sec, cot
- Signs of functions depend on quadrant location
- Fundamental identities: sin²θ + cos²θ = 1, tan²θ + 1 = sec²θ, cot²θ + 1 = csc²θ
- Even/odd functions: cos(-θ) = cos(θ), sin(-θ) = -sin(θ)
- Domain and range restrictions for each function
Definitions
Quadrant Signs
QI: all positive, QII: sin positive, QIII: tan positive, QIV: cos positive
Importance
Determines the sign of trigonometric function values
Reciprocal Functions
csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
Importance
Alternative ways to express trigonometric relationships
Common mistakes
- Forgetting to check quadrant when determining signs
- Confusing reciprocal relationships
- Not using identities to simplify complex expressions
Sum and Difference Formulas
Formulas
sin(A + B) = sin A cos B + cos A sin B
Example
sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
Variables
A, B = any angles
Application
Finding sine of angle sums
cos(A + B) = cos A cos B - sin A sin B
Example
cos 75° = cos(45° + 30°) = cos 45° cos 30° - sin 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4
Variables
A, B = any angles
Application
Finding cosine of angle sums
tan(A + B) = (tan A + tan B)/(1 - tan A tan B)
Example
tan 75° = tan(45° + 30°) = (tan 45° + tan 30°)/(1 - tan 45° tan 30°) = (1 + √3/3)/(1 - √3/3) = 2 + √3
Variables
A, B = angles where tangent is defined
Application
Finding tangent of angle sums
Exam tips
- Practice deriving exact values for angles like 15°, 75°, 105°
- Remember that difference formulas follow from sum formulas
- Use these formulas to verify trigonometric identities
Key points
- Addition formulas for sine, cosine, and tangent
- Subtraction formulas derived from addition formulas
- Applications in solving trigonometric equations
- Used to find exact values of non-special angles
- Foundation for double angle and half angle formulas
Definitions
Sum Formula
Formulas for trigonometric functions of angle sums (A + B)
Importance
Allows calculation of trigonometric values for non-standard angles
Difference Formula
Formulas for trigonometric functions of angle differences (A - B)
Importance
Derived from sum formulas by substituting -B for B
Common mistakes
- Mixing up addition and subtraction formulas
- Sign errors when applying formulas
- Not simplifying final answers completely
Laws of Sines and Cosines
Formulas
Law of Sines: a/sin A = b/sin B = c/sin C
Example
Triangle with A = 60°, B = 45°, a = 10. Find b: b/sin 45° = 10/sin 60°, so b = 10 sin 45°/sin 60° = 10(√2/2)/(√3/2) = 10√6/3
Variables
a, b, c = sides opposite to angles A, B, C respectively
Application
Solving triangles with known angles and sides
Law of Cosines: c² = a² + b² - 2ab cos C
Example
Triangle with a = 5, b = 7, C = 60°. Find c: c² = 5² + 7² - 2(5)(7)cos 60° = 25 + 49 - 70(1/2) = 74 - 35 = 39, so c = √39
Variables
a, b = two known sides, C = included angle, c = opposite side
Application
Finding third side when two sides and included angle are known
Exam tips
- Identify which law to use based on given information
- Draw and label triangles clearly
- Always check if your answer makes sense (angles sum to 180°)
Key points
- Law of Sines relates sides and angles in any triangle
- Law of Cosines generalizes Pythagorean theorem for any triangle
- Use Law of Sines when given two angles and one side, or two sides and an angle opposite one of them
- Use Law of Cosines when given three sides, or two sides and included angle
- Applications in solving oblique triangles and real-world problems
Definitions
Oblique Triangle
Any triangle that is not a right triangle
Importance
Requires Laws of Sines and Cosines since SOH-CAH-TOA doesn't apply
Ambiguous Case
When Law of Sines may yield two possible triangles
Importance
Occurs when given two sides and an angle opposite the shorter side
Common mistakes
- Using wrong law for given information
- Not checking for ambiguous cases in Law of Sines
- Forgetting that angles in triangle must sum to 180°
Connections
- Trigonometry connects to Geometry through triangle relationships and circle properties
- Applications in Physics for wave motion, oscillations, and vector analysis
- Foundation for Calculus concepts including derivatives and integrals of trigonometric functions
- Real-world applications in engineering, navigation, architecture, and astronomy
- Connects to Complex Numbers through Euler's formula and polar coordinates
Quick review questions
Convert 150° to radians.
Answer
5π/6 radians
Use formula: θ_rad = (π × θ_deg)/180° = (π × 150°)/180° = 5π/6
In a right triangle, if sin θ = 4/5, find cos θ.
Answer
cos θ = 3/5 (assuming θ is acute)
Using sin²θ + cos²θ = 1: cos²θ = 1 - (4/5)² = 1 - 16/25 = 9/25, so cos θ = ±3/5
Find the exact value of sin 15°.
Answer
(√6 - √2)/4
sin 15° = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4
In triangle ABC, a = 8, b = 6, C = 120°. Find c using Law of Cosines.
Answer
c = 2√19
c² = a² + b² - 2ab cos C = 8² + 6² - 2(8)(6)cos 120° = 64 + 36 - 96(-1/2) = 100 + 48 = 148, so c = √148 = 2√37
What is the reference angle for 240°?
Answer
60°
240° is in Quadrant III. Reference angle = 240° - 180° = 60°
Diagrams
Trigonometry Concept Overview
Comprehensive overview of all major trigonometry concepts showing relationships between topics
Triangle Problem Solving Decision Tree
Decision flowchart for determining which method to use when solving triangle problems
Converting Degrees to Radians Process
Step-by-step process for converting between degrees and radians
UPCAT Trigonometry Question Distribution
Typical distribution of trigonometry question types in UPCAT exams showing which topics to prioritize in review
Exam strategy
Focus on memorizing special angle values and key identities. Practice identifying which law or formula to use based on given information. Draw clear diagrams for every problem. Master the step-by-step approach: identify given information, choose appropriate method, substitute values carefully, and verify answers. For UPCAT preparation, emphasize exact values over decimal approximations, and practice both computational problems and conceptual questions about domains, ranges, and properties of trigonometric functions.