we fix an acute angle θ, then all right-angled triangles that have θ as one of their angles are similar. So, in all such triangles, corresponding pairs of sides are in the same ratio.Right angle triangle, sides labelled 'hypotenuse', 'opposite' and 'adjacent.The side opposite the right angle is called the hypotenuse. We label the side opposite θ as the opposite and the remaining side as the adjacent. Using these names we can list the FOLLOWING standard ratios:sinθ=oppositehypotenuse,cosθ=adjacenthypotenuse,tanθ=oppositeadjacent.Allied to these are the three reciprocal ratios, COSECANT, SECANT and COTANGENT:cosecθ=hypotenuseopposite,secθ=hypotenuseadjacent,cotθ=adjacentopposite.These are called the reciprocal ratios ascosecθ=1sinθ,secθ=1cosθ,cotθ=1tanθ.The trigonometric ratios can be used to find lengths and angles in right-angled triangles.ExampleFind the value of x in the following triangle.Right angle triangle, adjacent labelled as x, opposite marked as 3, angle created by the adjacent and hypotenuse marked as 72 degrees.SolutionWe havex3=cot72∘=1tan72∘.Hencex=3tan72∘≈0.97,correct to two decimal places.Special anglesThe trigonometric ratios of the angles 30∘, 45∘ and 60∘ can be easily expressed as FRACTIONS or surds, and students should commit these to memory.Trigonometric ratios of special anglesθ sinθ cosθ tanθ30∘ 12 3–√2 13–√45∘ 12–√ 12–√ 160∘ 3–√2 12 3–√Right angle triangle on the left, adjacent marked as 1, hypotenuse marked as root 2, on the right a Right angle triangle, adjacent marked as 1, hypotenuse marked as 2.Triangles for the trigonometric ratios of special angles.