Algebra and Trigonometry Quiz - Standard Form & Surds

Standard Form and Surds

• Standard Form: Express numbers as ( a \times 10^n ), where ( 1 \le a < 10 ) and ( n ) is an integer.

• Simplifying Surds: Combine square roots using these rules:

  • ( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} )
  • ( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} )

• Rationalizing the Denominator: Eliminate square roots from denominators using:

  • ( \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} )

Trigonometry

• Trigonometric Ratios (Right-angled Triangles):

  • Sine: ( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} )
  • Cosine: ( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} )
  • Tangent: ( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} )

• Exact Trigonometric Values:

  • ( \sin 30^\circ = \frac{1}{2} ), ( \sin 45^\circ = \frac{\sqrt{2}}{2} ), ( \sin 60^\circ = \frac{\sqrt{3}}{2} )
  • ( \cos 30^\circ = \frac{\sqrt{3}}{2} ), ( \cos 45^\circ = \frac{\sqrt{2}}{2} ), ( \cos 60^\circ = \frac{1}{2} )
  • ( \tan 30^\circ = \frac{1}{\sqrt{3}} ), ( \tan 45^\circ = 1 ), ( \tan 60^\circ = \sqrt{3} )

• Trigonometric Graphs (0° ≤ x ≤ 360°): The graphs of ( y = \sin x ), ( y = \cos x ), and ( y = \tan x ).

• Trigonometric Identities:

  • ( \sin^2 \theta + \cos^2 \theta = 1 )
  • ( \tan \theta = \frac{\sin \theta}{\cos \theta} )

• Sine Rule: ( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )

• Cosine Rule: ( c^2 = a^2 + b^2 - 2ab \cos C )

• Area of a Triangle: ( \text{Area} = \frac{1}{2}ab \sin C )

Elevation and Depression

• Angles of Elevation/Depression: Trigonometry is used to find distances or heights related to elevation/depression angles.

3D Trigonometry

• Perpendicular Distance: The shortest distance from a point to a line or plane in 3D space.

• Finding Angles in 3D: Apply trigonometry and Pythagorean Theorem for solving angle calculations.

Transformations

• Reflections: Flips across lines (e.g., ( y = x ), ( y = -x )).

• Rotations: Turns around a fixed point (e.g., multiples of 90°).

• Enlargements: Scaling with a scale factor (k).

• Translations: Sliding by a vector.

Vectors

• Vector Representation: A vector is represented as ( \mathbf{a} = \begin{pmatrix} x \ y \end{pmatrix} ).

• Vector Addition/Subtraction: Add/subtract corresponding components. ( \mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \ y_1 + y_2 \end{pmatrix} ) and ( \mathbf{a} - \mathbf{b} = \begin{pmatrix} x_1 - x_2 \ y_1 - y_2 \end{pmatrix} ).

• Scalar Multiplication: ( k \cdot \mathbf{a} = \begin{pmatrix} kx \ ky \end{pmatrix} )

• Magnitude: ( |\mathbf{a}| = \sqrt{x^2 + y^2} )

Vector Geometry

• Line Segments: Directed line segments between two points.
• Vector Sum/Difference: Combining or subtracting vectors in geometric situations.
• Scalar Multiplication: Changing the length of a vector.
• Applications: Solving problems involving direction, distance or speed using vectors.