Trigonometry
What is Trigonometry?
Ratios of the lengths of sides of a right-angled triangle relative to its angles (sine, cosine, tangent, etc.).
Key formula / rule: Pythagorean Identity
Key points
- Define and calculate the six trigonometric ratios.
- Understand and apply fundamental trigonometric identities.
- Solve problems involving angles of elevation and depression.
- Calculate heights and distances using trigonometry.
Common exam trap
Confusing sine and cosine definitions (opposite vs. adjacent).
Definitions
- Term
Trigonometric Ratios
- Meaning
Ratios of the lengths of sides of a right-angled triangle relative to its angles (sine, cosine, tangent, etc.).
- Term
Trigonometric Identity
- Meaning
An equation involving trigonometric functions that is true for all values of the variable for which the functions are defined.
- Term
Angle of Elevation
- Meaning
The angle formed by a horizontal line and the line of sight to an object above the horizontal line.
- Term
Angle of Depression
- Meaning
The angle formed by a horizontal line and the line of sight to an object below the horizontal line.
Learning objectives
Define and calculate the six trigonometric ratios.
Understand and apply fundamental trigonometric identities.
Solve problems involving angles of elevation and depression.
Calculate heights and distances using trigonometry.
Convert angles between degrees and radians.
Formulae
- Name
Pythagorean Identity
- Note
Fundamental identity relating sine and cosine.
- Expression
sin²θ + cos²θ = 1
- Name
Reciprocal Identities
- Note
Relates trigonometric functions to their reciprocals.
- Expression
cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
- Name
Ratio Identities
- Note
Expresses tangent and cotangent in terms of sine and cosine.
- Expression
tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
- Name
Trigonometric Ratios (Right Triangle)
- Note
Definitions for an acute angle θ in a right-angled triangle.
- Expression
sinθ = Opposite/Hypotenuse, cosθ = Adjacent/Hypotenuse, tanθ = Opposite/Adjacent
- Name
Angle of Elevation
- Note
Used in height and distance problems.
- Expression
Angle measured upwards from the horizontal.
- Name
Angle of Depression
- Note
Used in height and distance problems; alternate interior angle equals angle of elevation.
- Expression
Angle measured downwards from the horizontal.
Prerequisites
Basic Geometry (types of triangles, Pythagoras theorem).
Basic Algebra (solving equations).
Common mistakes
Confusing sine and cosine definitions (opposite vs. adjacent).
Incorrectly applying reciprocal or ratio identities.
Errors in calculating values for standard angles.
Assuming trigonometric relationships apply universally without considering the triangle type (e.g., right-angled).
Unit conversion errors between degrees and radians.
Keywords
Trigonometry
Trigonometric Ratios
Sine
Cosine
Tangent
Identities
Angles
Heights and Distances
Angle of Elevation
Angle of Depression
Radians
Degrees
Practice preview
What is the value of cos²(θ) + sin²(θ)?…
easy
If sin(θ) = 12/13, and θ is in the first quadrant, find the value of cos(θ).…
medium
What is the value of sin(30°)?…
easy