Sine and Cosine Rules (4.24.1) | OCR GCSE Maths (Higher) Notes

Understanding the sine and cosine rules is essential for solving problems involving non-right-angled triangles. These rules allow us to find unknown sides and angles in any triangle, expanding our problem-solving toolkit beyond the Pythagorean theorem, which applies only to right-angled triangles.

Introduction to the Sine Rule

The sine rule is a powerful tool for dealing with triangles where we don't have a right angle. It's especially useful when we have either two angles and one side (AAS or ASA) or two sides and a non-enclosed angle (SSA) information. The rule states:

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

where $a$ , $b$ , and $c$ are the lengths of the sides of the triangle, and $A$ , $B$ , and $C$ are the opposite angles.

Image courtesy of Cue Math

Example Using the Sine Rule

Given: $a = 7$ cm, $A = 35°$ , and $B = 60°$ , find $b$ .

Solution:

Metaphase

During metaphase, chromosomes align at the equatorial plate of the cell. The spindle fibres attach to the centromeres of each chromosome, ensuring that sister chromatids are positioned ready for separation. This stage is critical for ensuring each daughter cell receives an identical set of genetic material.

Chromosomes line up at the cell equator in a single row.
Spindle fibres from opposite poles attach to each centromere.
The cell checks that all chromosomes are correctly attached before proceeding.

Anaphase

Anaphase begins when the centromeres divide and sister chromatids are pulled apart toward opposite poles of the cell. Motor proteins within the spindle fibres generate the force required to shorten the fibres and move chromatids. This is one of the shortest stages of mitosis but essential for equal distribution of DNA.

Sister chromatids separate and move to opposite poles.
Spindle fibres shorten, drawing chromosomes apart.
Chromosome number remains the same until cytokinesis completes.

Telophase

In telophase, chromosomes arrive at the poles and begin to decondense. Nuclear envelopes re-form around each set of chromosomes, producing two distinct nuclei within the same cell. The spindle apparatus disassembles and the cell prepares for the final stage of division.

Cytokinesis

Cytokinesis is the physical separation of the cytoplasm into two daughter cells. In animal cells, a contractile ring of actin filaments pinches the cell membrane inward. In plant cells, a cell plate forms along the equator, eventually developing into a new cell wall between the two daughter cells.

Animal cells divide by cleavage furrow formation.
Plant cells divide by cell plate formation.
Each daughter cell enters interphase with a complete set of chromosomes.

Summary

Mitosis produces two genetically identical daughter cells from a single parent cell. The process is tightly regulated by checkpoints at each stage to prevent errors in chromosome segregation. Understanding mitosis is fundamental to topics including growth, tissue repair, and the development of cancer therapies.

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Written by:

Dr Rahil Sachak-Patwa

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.