COORDINATE GEOMETRY.
EQUATION OF A STRAIGHT LINE
SOLVED EXAMPLES.
1.

(

)

(

Solution.
(

Now, using the formula

) we have:

(
(

).

)

.
.

(

2.

)

Solution.
(

)

(

)

(
(

)

),

(

)

)

(

)

(

)

Solution.
(

)

(

)

(
)

(

)

EQUATION OF A CIRCLE.
The general equation of a circle is of the form

.Where (–
√

is the centre of the circle and the radius is:

Finding the equation of a circle of a circle given its radius and centre
Let (
) be any point on a circle whose centre is ( circle is given by : (
)
(
)

) and

(

r
(

(

)
)

)

0

If the centre is at the origin (
(
)
(
)

) then the equation becomes:

‘

the equation of a

)

Solved Examples.
1. Find the equation of the circle with centre (

) and radius

Solution.
Using (

)
(

,
(

(
)-

)

we have,

)
(

)

(

)

2. Find the equation of a circle with centre (

) which passes through the point(

Solution.
(

)
(

(

)

)

(

)

√
(

)

(

(

)

(

,

)
)

(√

)

3. Find the centre and radius of the circle
Solution.

( )

Comparing (1) with the general equation of a circle

The radius of the circle is:
√

The centre is (

√

√
)

(

)

)

EXERCISE.
1. Find the centre and radius of the circle

2. Find the centre and radius of the circle

3. Find the equation of the circle which passes through the points
(

)

(

)

(

)

TRIGONOMETRY.
Basic trigonometric ratios and their reciprocals

From the diagram above, the three (3) main trigonometric ratios are;

1.
2.
3.
Their reciprocals are:

1.
2.
3.

Trigonometric Ratios of Special Angles
From the triangle
√

√
√
√
√

5

Again, from the triangle PQR;

√

√

√
√
√

Basic Trigonometric identities
1.
2.
3.
4.
Compound angles identities.
(
)
1.
(
)
2.
(
)
3.
(
)
4.
(
)
5.

6.

(

)

Multiple angles identities.

1.
2.
3.
4.
5.
6.
7.
8.

Proofs

(

1.

(

)

(

)

(

(

)

(

)

)

2.

(

3.

)

(

)

4.
(

)

5.
(

6.

(

7.

(

)

)

)
(

)

(
(

8.

(
(

)
)

)
)

(
(

)
)

9.

(

)

.

/
(

)
(

)

SOLVED EXAMPLES.

1. Prove that
Solution.

.

/
.

/

2. Prove that √
Solution
√

√

(

)

√

√

(

)

√

Hence, √
3. Find the truth set of the equation
Solution.
But
(

(

*

)(

)

)

+

4. Find the truth set of the equation
Solution
But
(

)

Let

(

)(

)

But

*

Exercise
1. Show that
2. Find the truth set of the equation

+

DIFFERENTIATION.
Differentiation is the process of finding the derivative of a function. The rates of differentiation are as follows;
( )
( )
1.
( )
( )



( )

( )



2.



(

( )

( )

( )
( )
( )

3.

( )

4.

( )

( )

( )
( )

( )




( )
( )
( ) ( )

( )
( )

( )

( )

( )
( )

5.
( )

( )



( )

( )
( )

( )

(
( )

( )

( )

)(

)

( )

)
(

6.
( )
( )
( )
( )
( )
, ( )-

( )
( )



( )

( )
(

( )

)(
(

( )
( )

)
)

(

)

(
(

( )

)
)

(

)

( )

7.
( )

, ( )-

( )



( )
( )

(

, ( )(

( )

)

)
(
(


(

)
)
)

(

)

8. Higher order derivatives are found by successively applying the rules of the differentiation to the derivatives of the previous order
( )
( )
( )
( )
( )
( )

9.





(

)

10.

( )

( )

( )

( )




11.


( )



( )
(

)
(

1.

(

2.

√

3.

(

)

)(

)

)

4.
5.

( )

(

)(

)

( )

INTEGRATION.
Integration is the reverse of differentiation, it is sometime called antidifferentiation.
The rules of integration are as follows:

1.
∫

∫

2.
∫

3.

;
∫



∫



∫

4. Integral of
∫

∫

5. The integral of
∫

∫

6. The integral of
∫

∫

7. The integral of Sine;
∫

8. The integral of
∫

9. Integral of
∫

10. Integral of
∫

11. Integral of
∫

∫

Definite Integrals
1. ∫ (

)
(

)(

2. ∫ .

,

-

/

=∫ .

/

=∫ .
=0

∫ (

/

)

1

Integration by substitution
1. ∫

(

∫

)

(

)

∫
2. ∫

∫

(

)

(

√

∫

√

∫ √

∫
[

]

[ √
0√
,

]
√
-

1

)

Integration by Parts
1. ∫

Solution.

∫

∫
∫

∫

2. ∫

Solution.

∫

∫
∫

∫

(

)

EXERCISE.
1. Evaluate
∫

(

)

INTRODUCTION TO DIFFERENTIAL EQUATIONS
A differential equation is an equation involving an unknown function and its derivatives. The following are differential equations involving the unknown function .
1.
2.

( )
(

3.
4. .

/

)
. /

( )

5.







A differential equation is an ordinary differential equation (ODE) if the unknown function depends on only one independent variable. If the unknown function depends on two or more independent variables, the differential equations is a partial differential equation(PDE).Equations 1,2,3,and 4 are examples of ordinary differential equation since the unknown function ‘ ’ depends solely on the variable ’ .Equation five(5) is a partial differential equation, Since depends on both the independent variable ‘ and
.
The order of a differential equation is the highest derivative in the equation in the equation. Equation 1 is a first-order differential equation. Equation 2, 4 and 5 are second-order differential equations. Equation 3 is a third-order differential equation.
The degree of the differential equation is the power of the highest derivative in the equation. Equations 1, 2, 3 and 5 are of degree one (1) and equation 4 is of degree 3.
A solution of a differential equation is a function that satisfies the differential equation identically. Examples.
1. Determine whether or not ( )

( )

Hence,

(

2. Determine whether or not

)

(

)

( )

Solution.

Hence (

)

(

)

(

)

( )
3. Determine whether or not ( )
( )
( )
( )
(

)

( )



A particular Solution of a differential equation is any one solution.
The general solution of a differential equation is the set of all solutions.

EXERCISE.
1. State the order and degree of the following differential equations:
) (

)

)
)
)

( )
(

)

(

)

2. Which of the following function are solutions of the differential equation
( )

( )

( )

( )

( )