Additional Mathematical Formulas
Length of an arc (Radians)
l = rθ
(r = radius, θ = angle in radians)
Area of a sector (Radians)
A = 1/2 r2θ
(r = radius, θ = angle in radians)
Trigonometric Identities
cos2 θ + sin2 θ = 1
tan θ = sin θ / cos θ
Transformation matrices
[ cos 2θ sin 2θ ]
[ sin 2θ -cos 2θ ]
Reflection in the line y = (tan θ) x
[ k 0 ]
[ 0 1 ]
Horizontal stretch by scale factor of k
[ 1 0 ]
[ 0 k ]
Vertical stretch with scale factor of k
[ k 0 ]
[ 0 k ]
Enlargement with scale factor of k, centre (0, 0)
[ cos θ -sin θ ]
[ sin θ cos θ ]
Anticlockwise rotation of angle θ about the origin (θ > 0)
[ cos θ sin θ ]
[ -sin θ cos θ ]
Clockwise rotation of angle θ about the origin (θ > 0)
Magnitude of a vector
|v| = √(v12 + v22 + v32)
Vector equation of a line
r = a + λb
Parametric form of the equation of a line
x = x0 + λl
y = y0 + λm
z = z0 + λn
Scalar product (Dot product)
v · w = v1w1 + v2w2 + v3w3
v · w = |v||w|cosθ
where θ is the angle between v and w
Angle between two vectors
cosθ = (v1w1 + v2w2 + v3w3) / (|v||w|)
Vector product (Cross product)
v × w = [ v2w3 - v3w2 ]
[ v3w1 - v1w3 ]
[ v1w2 - v2w1 ]
|v × w| = |v||w|sinθ
where θ is the angle between v and w
Area of a parallelogram
A = |v × w|
Where v and w form two adjacent sides of a parallelogram
Geometry & Trigonometry: Understanding the Basics
What is the difference between Geometry and Trigonometry?
While both are branches of mathematics dealing with shapes and their properties, they have distinct focuses:
Geometry is the broader field. It studies the properties, relationships, and measurements of points, lines, angles, surfaces, solids, and higher-dimensional analogs. It's about understanding the nature of space and the shapes within it.
Trigonometry is a more specialized branch that specifically focuses on the relationships between the angles and side lengths of triangles. It utilizes trigonometric functions (like sine, cosine, and tangent) to perform calculations and solve problems, particularly those involving right-angled triangles. Trigonometry is often seen as a powerful set of tools used within geometry and other scientific fields.
In short: Geometry is about all shapes and spatial properties; Trigonometry is a toolkit for analyzing triangles.
Are Geometry and Trigonometry the same?
No, Geometry and Trigonometry are not the same, but they are very closely related and often taught in sequence.
Think of it this way:
- Geometry provides the foundational understanding of shapes, including triangles, angles, and their properties (like the Pythagorean theorem).
- Trigonometry builds upon this geometric foundation by introducing specific functions (sine, cosine, tangent) and techniques to calculate unknown side lengths or angles within triangles. It's an extension or a specialized application of geometric principles.
So, while distinct, a good understanding of geometry is typically essential before diving into trigonometry, as trigonometry uses geometric concepts as its starting point.
