3D Trigonometry - Higher
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Statistics
Sets and Venn diagrams
Sampling and bias
Collecting data: types and classes of data
Mean, median, mode and range
Simple charts and graphs
Pie charts
Scatter graphs
Frequency tables: finding averages
Grouped frequency tables
Box plots - Higher
Cumulative frequency - Higher
Histograms and frequency density - Higher
Interpreting data
Comparing data sets
Probability
Trigonometry
Angles and geometry
Shapes and area
Properties of 2D shapes
Congruence: conditions for congruent triangles
Similar shapes: Scaling
The four transformations
Area and perimeter: Formulae
Area and circumference of circles: Formulae
3D shapes: faces, edges, vertices
Surface area of 3D shapes: Nets, formulae
Volume of 3D shapes: Formulae
Volume of 3D shapes: Comparing, rates of flow
Area and volume scale factors
Projections and elevations of 3D shapes
Ratio proportion and rates of change
Graphs
Coordinates and midpoints
Straight line graphs
Drawing straight line graphs
Finding the gradient of a straight line
Equation of a straight line: y = mx + c
Coordinates and ratio
Parallel and perpendicular lines
Quadratic graphs
Reciprocal and cubic graphs
Exponential graphs and circles - Higher
Trigonometric graphs - Higher
Solving equations using graphs
Graph transformations - Higher
Real-life graphs
Distance-time graphs
Velocity-time graphs - Higher
Gradients of real-life graphs - Higher
Algebra
Simplifying algebraic expressions
Multiplying and dividing algebraic expressions
Single brackets: Expanding and factorising
Double brackets: Expanding and factorising
Double and triple brackets - Higher
Solving equations
Expressions, equations, formulae, functions and identities
Writing formulae and equations from word problems
Writing formulae and equations from diagrams
Rearranging formulae
Factorising quadratics
The quadratic formula - Higher
Complete the square - Higher
Algebraic fractions - Higher
Sequences
Finding the nth term
Solving inequalities
Inequalities on graphs - Higher
Iteration - Higher
Simultaneous equations: elimination and substitution
Non-linear simultaneous equations - Higher
Algebraic proof - Higher
Composite and inverse functions - Higher
Number
Types of numbers
Order of operations: BODMAS
Multiplying and dividing by powers of 10
Multiplying and dividing whole numbers
Multiplying and dividing decimals
Negative numbers: add, subtract, multiply, divide
Prime numbers and prime factorisation
Multiples, factors and prime factors
LCM and HCF
Fractions
Fractions, decimals and percentages
Writing recurring decimals as fractions
Rounding: Integers, decimal places, significant figures
Estimation
Error intervals
Upper and lower bounds - Higher
Powers and roots: Square and cube numbers
Laws of indices: multiply, divide, brackets
Index laws: negative and fractional indices - Higher
Surds: Simplify, add and subtract - Higher
Rationalising surds - Higher
Standard form calculations
Explainer Video
Summary
3D Trigonometry
In a nutshell
D trigonometry is very similar to D trigonometry. The key difference is being able to process a D problem into a D one to make it easier to visualise and solve.
Planes and lines in D
In D trigonometry, you need to know what a plane is and be able to find angles between lines and planes.
Definition: A plane is a flat, two dimensional surface.
Converting D problems to D problems
To convert D trigonometry problems to D ones, you have to be able to identify exactly what angles and sides you need, then draw a D shape that simplifies the question. It is best shown with an example.
Example 1
Give a D representation of the angle between the line and the plane in this cuboid.
First, identify the line and the plane . The line is the diagonal of the cuboid. The plane is the bottom rectangle of the cuboid. To identify the angle between the line and place, you need to draw a triangle.
One of the sides of this triangle is . Another one of the sides will be on the plane . To keep this two dimensional, use the line as it is in the same direction as . To complete the triangle, the third line has to be . Here is an illustration:
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The transformation from a 3D drawing to a 2D one simplifies the problem and makes it much easier to solve.
Solving 3D trigonometry problems
To solve a 3D trigonometry problem, you first have to convert it to a 2D one and then solve like normal.
Example 2
In the cuboid from the previous example, the sides , , and are , , respectively. Find the angle between the line and the plane to decimal place.
The above example already transformed the problem to a 2 dimensional one. Now, you can use SOHCAHTOA to find the missing angle.
However, there is not enough information to find the angle - at least 2 sides are needed. Therefore, use Pythagoras to find the side by focusing on the triangle :
Substitute in :
Now, use trigonometric ratios to find the desired angle - call it for simplicity:
FAQs - Frequently Asked Questions
How do we transform a 3D trigonometric problem into a 2D one?
Identify the lines and angles that are absolutely necessary for your problem and try to make a 2D shape out of them.
How do you solve a 3D trigonometry problem?
First transform the problem into a 2D one and solve as you would a standard 2D trigonometry problem.