Trigonometry

The word "Trigonometry" is derived from the Greek word "Tri-Gonia-Metron" where 'Tri' means 'three', 'Gonia' means 'angles' and 'Metron' mean 'measure'. So, trigonometry is a branch of mathematics which concerned with the measurement of sides, angles and their relation to a triangle.

Trigonometric Ratios

The word trigonometry comes from the combination of the word triangle goes on having the meaning of the measurement of three angles of triangles. Hence, this is also known as the measure of the triangle. Trigonometry has wide application in the field of mathematics and science. With the help of right angles triangle, trigonometric ratios of an angle are found. Without the help of trigonometric ratios, both the development and expansion of physics and also of engineering are impossible. Hence, trigonometry is the very important branch of mathematics.

Above figure shows the shadow of the poles formed at 3 pm that stand perpendicularly on the road. For each figure, the ratio of height of pole and length of shadow and height: length are tabulated below:

Hence, the height of every pole and the length of their shadows are in proportion. The angle made by the top of the pole and with the top of shadow on the ground is also equal.

Fundamentals of Trigonometric Ratios

The ratio of any two sides of a right-angled triangle taking one of the side as reference are the fundamentals of trigonometric ratios.

Let know on detail about ratio with a figure. Here, in the given figure, ΔABC is a right angled triangle. \(\angle\)B = 90^oand \(\angle\)C =θ.

Let's take\(\angle\)C as a reference angle

The opposite side of angle C (perpendicular) (P) = AB

The adjacent side of angle C (base) (B) = BC and (hypotenuse) (H) = AC

We can make three relation with the reference angle.

• relation between perpendicular and hypotenuse

In the above figure, the ratio of AB (perpendicular) to AC (base) with reference angle is called sine θ.

∴ sin θ = \(\frac{AB}{AC}\) = \(\frac{perpendicular}{hypotenuse}\) = \(\frac{p}{h}\)

• relation between base and hypotenuse

In the above figure, the ratio of BC (base) to AC (base) with reference angle is called cosine θ.

∴ cos θ = \(\frac{BC}{AC}\) = \(\frac{base}{hypotenuse}\) = \(\frac{b}{h}\)

• relation between perpendicular and base

In the above figure, the ratio of AB (perpendicular) to BC (base) with reference angle is called tangent θ.

∴ tan θ = \(\frac{AB}{BC}\) = \(\frac{perpendicular}{base}\) = \(\frac{p}{b}\)

Pythagoras Theorem

The relationship between the three sides of a triangle is simply known as Pythagoras Theorem. The relation was given by the popular Mathematician Pythagoras so it is called as Pythagorean theorem.
In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
According to this theorem "In any right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of squares of perpendicular and base".

By Pythagoras Theorem

Hypotenuse (h²) = Perpendicular (p²) + Base (b²)
or, h²= p²+ b²
From this theory we can derive,
h = \(\sqrt{p^{2} + b^{2}}\)
p = \(\sqrt{h^{2} - b^{2}}\)
b = \(\sqrt{h^{2} - p^{2}}\)

Theoretical proof:

Given: ΔABC is a right angled triangle in which \(\angle\)ABC = 90^o.

To prove: CA² = AB² + BC²

Construction: BD ⊥AC is draawn

Proof: