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Trigonometric Ratios

From Pythagoras’ theorem, we know that the longest side of a right-angled triangle is called the hypotenuse. The other two sides of the triangle are named with reference to the acute angle in the triangle.

 

In the right-angled triangle ABC below, the angle of reference is ‘C’. The side opposite to the angle C is AB, called the opposite side. The side next to angle C is BC, called the adjacent side. The longest side, called the hypotenuse (here AC), is always opposite to the 90°.

sides in a triangle

 

Trigonometric ratios

There are three main trigonometric ratios. For every acute angle in a right-angle triangle, there is a particular value for its  {opposite}/{hypotenuse}, {adjacent}/{hypotenuse} and {opposite}/{adjacent} ratio that always stays the same (constant).

 

These ratios are called sine (written sin in short), cosine (written cos in short) and tangent (tan in short) respectively. For a given acute angle θ (theta),

sin θ = {opposite}/{hypotenuse}

cos θ ={adjacent}/{hypotenuse}

tan θ = {opposite}/{adjacent}

For example, sin 30° means the sine ratio for an angle of 30°. Similarly for cos and tan ratios.

 

Let’s now look at a right-angled triangle with opposite side = 5, adjacent side = 12 and hypotenuse = 13.

finding the angle given sides of a triangle

sin θ = {opposite}/{hypotenuse}~ = ~5/13~=~0.385

cos θ = {adjacent}/{hypotenuse}~ = ~12/13~=~0.923

tan θ = {opposite}/{adjacent}~ = ~5/12~=~0.417

Now let us look at another right-angled triangle with opposite side = 5 and adjacent side = 8. In order to find the sin, cos and tan for the triangle, we need the dimensions of hypotenuse.

finding the angle given sides of a triangle

Using Pythagoras’ rule a2 = b2 + c2, we find hypotenuse^2~=~8^2~+~5^2. So hypotenuse =  sqrt 89 . Now

sin θ = {opposite}/{hypotenuse}~ = ~5/sqrt 89~=~0.530

cos θ = {adjacent}/{hypotenuse}~ = ~8/sqrt 89~=~0.848

tan θ = {opposite}/{adjacent}~ = ~5/8~=~0.625