Sine, cosine and tangent in right triangles
In a right triangle, the ratio of two sides depends only on the angle. This gives sine, cosine and tangent, with examples such as ramps and ladders.
Mathematics – grade 10
Grade 10: trigonometry, circles and solids at a higher level, trigonometric and exponential functions, logarithms and similarity. The key ideas number and operation and data and chance have no topics of their own for grade 10 in the distribution table. The numbers are our own.
Unit 2Quantities and measurement
Trigonometry, circles and sectors, and further calculations with solids.
2.1trigonometry
Sine, cosine and tangent in right triangles and on the unit circle, the sine rule and the cosine rule.
Law of sines and law of cosines
Calculate sides and angles in any triangle with the law of sines and the law of cosines. A helper shows which law fits your problem.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
2.2calculations with circles and sectors
Arc length and area of sectors and composite figures, radians.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
2.3further calculations with solids
Calculations with solids at a higher level, including composite solids with cones or spheres.
Pyramid, cone and sphere
Pour three times and the prism is full: a pyramid holds exactly one third. Cones and spheres can be turned and measured in 3D.
Unit 3Structures and functional relationships
Trigonometric functions, exponential functions, exponential equations and logarithms; deepening the topics marked *.
3.1trigonometric functions
Periodic processes, projection on the unit circle, radians and the meaning of the parameters a, b, c.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
3.2exponential functions
Exponential growth, monotonicity, doubling time and half-life, asymptotic behaviour and parameters.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
Limits and asymptotes
What happens to a graph when x gets very large? A point runs outwards and a magnifier shows how close it gets to the asymptote.
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
3.3exponential equations
Solving exponential equations by trial, graphically and with logarithms.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
Logarithms as the inverse of exponentiation
The logarithm answers the question: to what power do I raise the base to get this number? With graphs, rules and exponential equations.
3.4logarithms
Understanding and using the logarithm as the inverse of taking powers.
Logarithms as the inverse of exponentiation
The logarithm answers the question: to what power do I raise the base to get this number? With graphs, rules and exponential equations.
3.5Deepening: quadratic functions and equations*
The topics from grades 7–9 marked * can be deepened in grade 10.
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
Unit 4Space and shape
Similarity of figures and the similarity criterion for triangles.
4.1similarity
Similar figures and the similarity criterion for triangles; the effect of the scale factor on lengths, areas and volumes.
Similar figures: how do area and volume grow?
Enlarge lengths, areas and solids. Doubling the length makes the area four times and the volume eight times as large.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Fachanforderungen, Gymnasium, Mathematics, grade 10