Any triangle has three sides and three angles. But Is there any relation between the sides and angles of a triangle?
Trigonometry aims to answer this question.
Before entering into its terms, let's revise Pythagoras theorem. It states
" In a right-angled triangle, the sum of squares of the (lengths of) sides of the right angle equals the square of (length of) the third side of the hypotenuse"
In the above figure, ∆ABC is a right-angled triangle with right angle at vertex B.
AB, BC are the sides of the right angle.
AC is the third side (of ∆ABC) or the hypotenuse.
By Pythagoras theorem
AB² + BC² = AC²
Please note I am not getting into proof of the theorem.
The first concept of trigonometry is trigonometric ratios (of angles)
Reviewing the above picture
In right triangle ABC angles A & C are non-right angles. [These angles are complementary to each other and vary between 0° and 90°]
∠A (or ∠C) have an opposite side BC (or AB), an adjacent side (other than hypotenuse) AB (or BC).
AC is the hypotenuse
There are six trigonometric ratios for an angle. sine, cosine, tangent, co-tangent, cosecant, secant.
Trigonometric ratios are only defined for a right-angled triangle
I shall list down the six trigonometric ratios of ∠A. Write yourself the trigonometric ratios of ∠C. I'd leave a hint though at the bottom. The sides and hypotenuse mentioned in the below list are the lengths respectively. The trigonometric ratios are numerical values (without any units)
- Sin A = opposite side of ∠A / hypotenuse
- Cos A = adjacent side of ∠A / hypotenuse
- tan A = opposite side of ∠A / adjacent side of ∠A
- cot A = adjacent side of ∠A / opposite side of ∠A
- cosec A = hypotenuse / opposite side of ∠A
- sec A = hypotenuse / adjacent side of ∠A
The first two ratios are sufficient to extract the other four ratios. Observe carefully the above list, try to extract the bottom four ratios from the top two. Check or find the solution below.
- Sin A = opposite side of ∠A / hypotenuse
- Cos A = adjacent side of ∠A / hypotenuse
- tan A = Sin A / Cos A
- cot A = ( tan A ) ^{-1}
- cosec A = ( sin A ) ^{-1}
- sec A= ( cos A ) ^{-1}
Now, try yourself for the trigonometric ratios of ∠C.
The hint goes as follows
Notice carefully that the adjacent side of ∠A is the opposite side of ∠C. Also, the opposite side of ∠A is the adjacent side of ∠C. We can note two results from this observation
- Sine (or cosine) of one angle (not the right angle) will be equal to cosine (or sine) of another angle (not the right angle)
- Sine (or cosine) of an angle will be equal to cosine (or sine) of its complementary angle
That's the first concept of trigonometry. We shall end here for today. Tomorrow, we shall enter the next concept, namely, trigonometric identities