Using the adjoining figure, calculate the values of :

(i) cos θ

(ii) tan Φ

(iii) cosec Φ

In right angled triangle ABC,

Hypotenuse = AC = 13 units

Perpendicular = BC = 5 units

By pythagoras theorem,

Base² = Hypotenuse² - Perpendicular²

Base² = 13² - 5²

Base² = 169 - 25

Base² = 144

Base = 12 units

AB = 12 units.

Draw a perpendicular CE on AD.

In triangle CED,

CE = AB = 12 units and AE = BC = 5 units

From figure,

AD = DE + AE

DE = AD - AE = 14 - 5 = 9 units

In Triangle CED,

By pythagoras theorem,

CD² = CE² + ED²

CD² = 12² + 9²

CD² = 144 + 81

CD² = 225

CD = $\sqrt{225}$ = 15 units.

(i) cos θ = $\dfrac{\text{base}}{\text{hypotenuse}}= \dfrac{AB}{AC} = \dfrac{12}{13}$.

(ii) tan Φ = $\dfrac{\text{perpendicular}}{\text{base}}= \dfrac{CE}{DE} = \dfrac{12}{9} = \dfrac{4}{3}$.

(iii) cosec Φ = $\dfrac{\text{hypotenuse}}{\text{perpendicular}} = \dfrac{CD}{CE} = \dfrac{15}{12} = \dfrac{5}{4}$.