Level 2 Further Mathematics AQA
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74 topics in 7 modules
☑️ Number 3 topics
- Use of Numbers
- The Product Rule for Counting
- Manipulation of Surds, including Rationalising the Denominator
☑️ Trigonometry 17 topics
- Understanding Sine, Cosine, and Tangent Ratios
- Experimenting with Unit Circle Definitions
- Applying Trigonometric Ratios to Find Missing Sides of Right-angled Triangles
- Employing the Sine and Cosine Rule
- Graphing Sine, Cosine and Tangent Functions
- Recognizing and Working with Co-functions and Reciprocal Functions
- Solving Basic and Advance Trigonometric Equations
- Recognizing and Utilizing Pythagorean Identities
- Exploring and Applying Double Angle Formulas
- Understanding and Working with Compound Angle Formulas
- Proving and Applying Trigonometric Identities
- Application of Trigonometry in Real Life (Word Problems and Contextual Applications)
- Understanding of Radians and Degrees Conversion
- Trigonometric Graph Transformations and Their Properties
- Identifying the Amplitude, Period, and Phase Shift of Trigonometric Functions from Graphs
- Working with the Trigonometric Form of Complex Numbers
- Applying Euler's Formula Involving Complex Numbers and Trigonometry.
☑️ Algebra 23 topics
- The Basic Processes of Algebra
- Definition of a Function
- Domain and Range of a Function
- Composite Functions
- Inverse Functions
- Expanding Brackets and Collecting Like Terms
- Expand (a + b)^n for Positive Integer n
- Factorising
- Manipulating of Rational Expressions
- Manipulation of Formulae and Expressions
- Use of the Factor Theorem for Rational Values of the Variable for Polynomials
- Completing the Square
- Drawing and Sketching of Functions
- Interpretation of Graphs
- Solution of Linear and Quadratic Equations
- Algebraic and Graphical Solution of Simultaneous Equations in 2 Unknowns
- Algebraic Solution of Linear Equations in 3 Unknowns
- Solution of Linear and Quadratic Inequalities
- Index Laws
- Algebraic Proof
- Using n-th Terms of Linear Sequences
- n-th Terms of Linear Sequences
- n-th Terms of Quadratic Sequences
☑️ Coordinate Geometry (2 Dimensions Only) 11 topics
- The Straight Line: Know and Use the Definition of a Gradient
- Know the Relationship between the Gradients of Parallel and Perpendicular Lines
- The Straight Line: Use Pythagoras' Theorem to Calculate Distance between 2 Points
- Use Ratio to Find the Coordinates of a Point on a Line Given the Coordinates of 2 Other Points
- The Equation of a Straight Line (y = mx + c)
- Draw a Straight Line from Given Information
- The Coordinate Geometry of Circles
- The Straight Line: Equation of a Circle
- The Coordinate Geometry of Circles: x^2 + y^2 = r^2
- The Coordinate Geometry of Circles: (x - a)^2 + (y - b)^2 = r^2
- The Equation of a Tangent at a Point on a Circle
☑️ Calculus 8 topics
- Differentiation: Gradient Function
- Differentiation of kx^n
- The Equation of a Tangent and Normal at any Point on a Curve
- Differentiation: Increasing and Decreasing Functions
- Understand and Use the Notation d^2y / dx^2
- Use of Differentiation to Find Maxima and Minima Points on a Curve
- Using Calculus to Find Maxima and Minima in Simple Problems
- Sketch or Interpret a Curve with Known Maximum and Minimum Points
☑️ Matrix Transformations 4 topics
- Multiplication of Matrices
- The Identity Matrix
- Transformations of the Unit Square in the x - y Plane
- Combination of Transformations
☑️ Geometry 8 topics
- Geometric Proof
- Trigonometry in Triangles
- Pythagoras' Theorem
- Sketch and Use Graphs y=sin x, y=cos x, y=tan x
- Using the Definitions sin θ, cos θ and tan θ
- Knowledge and use of 30-60-90 and 45-45-90 Triangles
- Know and use tanθ = sinθ / cosθ = and sin^2 θ + cos^2 θ = 1
- Solution of Simple Trigonometric Equations in Given Intervals
Level 2 Further Mathematics AQA Revision Content
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Level 2 Further Mathematics AQA - Trigonometry - Understanding Sine, Cosine, and Tangent Ratios Content Preview
Trigonometry
Understanding Sine, Cosine, and Tangent Ratios
Understanding Sine, Cosine, and Tangent Ratios
Introduction to Ratios in Trigonometry
- Trigonometry is a branch of mathematics that studies triangles, specifically the relationships among their angles and sides.
- The three primary ratios in trigonometry are sine (sin), cosine (cos), and tangent (tan).
- These ratios are built upon using the right-angled triangle, where one angle is exactly 90 degrees.
Sine Ratio
- The sine of an angle is described as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
- Mathematical representation: sin(θ) = Opposite/Hypotenuse.
- Remember, the hypotenuse is the longest side of the right-angled triangle, always opposite the right angle.
Cosine Ratio
- The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
- Mathematical representation: cos(θ) = Adjacent/Hypotenuse.
- Relevant to remember: the adjacent side is the side that forms the angle θ along with the hypotenuse.
Tangent Ratio
- The tangent of an angle is the ratio of the sine of the angle to the cosine of the angle. For a right-angled triangle, it's the ratio of the length of the opposite side to the length of the adjacent side.
- Mathematical representation: tan(θ) = sin(θ)/cos(θ) or tan(θ) = Opposite/Adjacent.
- Note this importance: the tangent ratio only makes sense as long as the cosine of the angle isn't zero to avoid division by zero.
Usage and Practical Applications
- These ratios are essential in countless situations where angles and distances need to be determined, such as in navigation, architecture, physics, engineering, and computer graphics.
- Calculators often have specific buttons for these trigonometric functions, enabling the computation of the ratios for any given angle.
- Sine and cosine ratios are crucial in a periodic phenomenon like sound and light waves, while tangent ratios predominantly find their use in slope calculations.
Remember SOHCAHTOA
- An easy way to remember the three basic functions sin, cos, and tan is by the mnemonic SOHCAHTOA.
- 'S' stands for Sine: Opposite/Hypotenuse.
- 'C' stands for Cosine: Adjacent/Hypotenuse.
- 'T' stands for Tangent: Opposite/Adjacent.
- This acronym is typically one of the most beneficial memorisation aids when first learning about trigonometric ratios.
Question: In a right-angled triangle, the hypotenuse is 13 cm and the side adjacent to angle \( \theta \) is 5 cm; write down the exact value of \( \cos(\theta) \).
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