tan (A + B)

Set 105º = 60º + 45º.

tan (60º + 45º)
= (tan 60º + tan 45º)/(1 - tan 60º tan 45º)

To find these tangent values,

draw a 30-60-90 triangle
whose sides are 1, √3, 2,

and a 45-45-90 triangle
whose sides are 1, 1, √2.

tan 60º

Tangent is TOA:
Tangent,
Opposite side (√3),
Adjacent side (1).

So tan 60º = √3/1 = √3.

Write +.

tan 45º

Tangent is TOA:
Tangent,
Opposite side (1),
Adjacent side (1).

So tan 45º = 1/1 = 1.

Write 1 -.

tan 60º = √3

So write √3.

tan 45º = 1

So write 1.

So (tan 60º + tan 45º)/(1 - tan 60º tan 45º)
= (√3 + 1)/(1 - √3⋅1).

1 - √3⋅1
= 1 - √3
= -(√3 - 1)

Write the minus sign
in front of the fraction.

To rationalize the denominator (√3 - 1),
multiply its conjugate (√3 + 1)
to both of the numerator and the denominator.

(√3 + 1)(√3 + 1)
= (√3 + 1)²
= 3 + 2⋅√3⋅1 + 1

Square of a Sum: (a + b)²

(√3 - 1)(√3 + 1)
= 3 - 1

Product of a Sum and a Difference: (a + b)(a - b)

3 + 2⋅√3⋅1 + 1 = 4 + 2√3

3 - 1 = 2

4/(-2) = -2
+2√3/(-2) = -√3

So
-2 - √3
is the answer.

Draw y = 2x
on a coordinate plane.

It says
y = mx is the image
that is rotated 45º counterclockwise about the origin.

So draw y = mx like this.

Set the central angle of y = 2x ∠A.

Tangent means the slope.
And the slope of y = 2x is 2.

So tan A = 2.

Find tan 45º.

Draw a 45-45-90 triangle
whose sides are 1, 1, √2.

Tangent is TOA:
Tangent,
Opposite side (1),
Adjacent side (1).

So tan 45º = 1/1 = 1.

The central angle of y = mx is
A + 45º.

So m = tan (A + 45º).

tan A = 2
tan 45º = 1

So tan (A + 45º)
= (2 + 1) / (1 - 2⋅1).

2 + 1 = 3

1 - 2⋅1 = 1 - 2

1 - 2 = -1

3/(-1) = -3

So m = -3.