Geometry and Trigonometry

The sine, cosine, and tangent ratios are fundamental in trigonometry for solving problems involving right-angled triangles. The sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. These ratios can be used to find unknown sides and angles in right-angled triangles. In non-right-angled triangles, the sine and cosine rules are invaluable. The sine rule relates the sides of a triangle to the sines of their opposite angles, allowing us to find unknown sides or angles. The cosine rule, which relates the lengths of the sides of a triangle to the cosine of one of its angles, is useful for finding an unknown side or calculating the angles when two sides and the included angle are known. The area of any triangle can be found using the sine formula, which is particularly useful when the height of the triangle is not known.

Example of Sine Ratio:

For a right-angled triangle with an angle \( A \), if the opposite side is \( o \) and the hypotenuse is \( h \), the sine ratio is given by:

$$ \sin(A) = \frac{o}{h} $$

Example of Cosine Ratio:

For the same triangle, the cosine ratio for angle \( A \), with adjacent side \( a \) and hypotenuse \( h \), is:

$$ \cos(A) = \frac{a}{h} $$

Example of Tangent Ratio:

The tangent ratio for angle \( A \), with opposite side \( o \) and adjacent side \( a \), is:

$$ \tan(A) = \frac{o}{a} $$

Using the Sine Rule:

For any triangle with sides \( a, b, \) and \( c \), and opposite angles \( A, B, \) and \( C \) respectively, the sine rule is expressed as:

$$ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} $$

Using the Cosine Rule:

To find an unknown side \( c \), given sides \( a \) and \( b \), and the included angle \( C \):

$$ c^2 = a^2 + b^2 - 2ab\cos(C) $$

Or to find an angle \( C \), given all three sides:

$$ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} $$

Area of a Triangle using Sine:

To find the area \( A \) of any triangle given two sides \( a, b \) and the included angle \( C \):

$$ A = \frac{1}{2}ab\sin(C) $$