Everything Maths Grade 10

Step 2: Use the theorem of Pythagoras to calculatey

r^2 =x^2 +y^2

y^2 =r^2 x^2

= (13)^2 (5)^2

= 169 25

= 144

y=± 12

Given thatlies in the third quadrant,ymust be negative. Thereforey= 12

Step 3: Substitute values forx,yandrand simplify

x= 5 ,y= 12 andr= 13.

LHS

tan^2 + 1 =

(y

x

) 2

+ 1

=

(

12

5

) 2

+ 1

=

(

144

25

)

+ 1

=

144 + 25

25

=

169

25

RHS

sec^2 =

(r

x

) 2

=

(

13

5

) 2

=

169

25

Therefore the LHS = RHS and we have proved thattan^2 + 1 =sec^2 .

NOTE:

Whenever you have to solve trigonometric problems without a calculator, it can be very helpful to make a

sketch.

Exercise 5 – 7:

1.Bis a point in the Cartesian plane. Determine without using a calculator:

0

y

x

B(1;3)

(^) X
a) OB b)cos c) cosec d)tan
136 5.8. Defining ratios in the Cartesian plane