Question

Given that Cos A=513and angle A is acute. Find the value of 2tan A + 3sin A without using a calculators.

Answer 1

`(492)/(65)`

Explanation:

Using the trigonometric identity

sin²A + cos²A = 1 , then

sinA =
`√(1-cos^2A)` thus

sinA =
`√(1-(5/13)^2)`

=
`\sqrt{1-(25)/(169) }`

=
`\sqrt{(144)/(169) }` =
`(12)/(13)` , then

tanA =
`(sinA)/(cosA)` =
`((12)/(13) )/((5)/(13) )` =
`(12)/(5)`

Thus

2tanA + 3sinA

= 2 ×
`(12)/(5)` + ( 3 ×
`(12)/(13)` )

=
`(24)/(5)` +
`(36)/(13)`

=
`(312)/(65)` +
`(180)/(65)`

=
`(492)/(65)`

Answer 2

This implies that

x2+2ax=4x−4a−13

or

x2+2ax−4x+4a+13=0

or

x2+(2a−4)x+(4a+13)=0

Since the equation has just one solution instead of the usual two distinct solutions, then the two solutions must be same i.e. discriminant = 0.

Hence we get that

(2a−4)2=4⋅1⋅(4a+13)

or

4a2−16a+16=16a+52

or

4a2−32a−36=0

or

a2−8a−9=0

or

(a−9)(a+1)=0

So the values of a are −1 and 9.

Explanation: