Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and side lengths of triangles. It is widely used in geometry, astronomy, physics, engineering, and navigation, and provides essential tools for modeling periodic phenomena.

Pythagorean Theorem

One of the fundamentals of Trigonometry is Pythagorean theorem (or Pythagoras' theorem), named after ancient Greek philosopher and polymath Pythagoras. Pythagorean Theorem works only when one of the inner angles of the triangle is exactly 90 degrees (right angle). Pythagorean theorem then states that in this case, the side not touching the right angle, also said to be directly across from the right angle, which is named the hypotenuse, is equal to the square root of the sum of the squared side lengths. It is commonly rewritten as A² + B² = C², where A and B are the lengths of the lines that create the right angle, and C is the hypotenuse.

Sine and Cosine

Regularly denoted as sin and cos, Sine and Cosine are trigonometric ratios.

Trigonometric Definition

sin(θ) = opposite / hypotenuse (often remembered as SOH)

cos(θ) = adjacent / hypotenuse (often remembered as CAH)

Tangent

Regularly denoted as tan, Tangent is equal to sinθ / cosθ.

Trigonometric Definition

tan(θ) = sin(θ) / cos(θ) = (opposite / hypotenuse) / (adjacent / hypotenuse) = opposite / adjacent (often remembered as TOA)

Cosecant, Secant, and Cotangent

Regularly denoted as csc, sec, and cot, Cosecant, Secant, and Cotangent are the reciprocal trigonometric ratios for the basic 3 trigonometric ratios Sine, Cosine, and Tangent.

Trigonometric Definition

csc(θ) = 1 / sin(θ) = 1 / (opposite / hypotenuse) = hypotenuse / opposite

sec(θ) = 1 / cos(θ) = 1 / (adjacent / hypotenuse) = hypotenuse / adjacent

cot(θ) = 1 / tan(θ) = 1 / (opposite / adjacent) = adjacent / opposite

Cosine and Sine rule(s)

The sine and cosine rules are mathematical relationships used to determine unknown sides and angles in non-right-angled (90°)triangles. They are extensions of the basic trigonometric ratios, sine and cosine.They are commonly used in geometry, surveying, navigation, physics, and engineering.

Sine Rule

The sine rule, also known as the law of sines, is a formula used to find unknown sides or angles in a triangle. It is particularly useful when you know a side and its opposite angle (two angles and one side (AAS/ASA), two sides and an angle opposite one (SSA)).

For a triangle with sides a,b,c opposite angles A,B,C:

a/sinA​=b/sinB​=c/sinC

It can also be written as:

sinA/a​=sinB/b​=sinC/c

Cosine Rule

The cosine rule, also called the law of cosines, relates the three sides of a triangle to one of its angles. It is particularly useful when you know two sides and the included angle (SAS) or all three sides (SSS).

For a triangle with sides a,b,c and angle C opposite side c:

c^2=a^2+b^2−2abcosC

Similarly:

a^2=b^2+c^2−2bccosA

b^2=a^2+c^2−2accosB

Cosine rule and Pythagoras' theorem

The cosine rule is a generalisation of Pythagoras' theorem:

If C=90° then:

c^2=a^2+b^2−2abcos90 Since: cos90°=0

the equation becomes:

c^2=a^2+b^2 which is Pythagoras' theorem.