sin (A - B)
How to find sin (A - B) by using its formula: formula, 1 example, and its solution.
Formula
Formula
sin (A - B) = sin A cos B - cos A sin B
For sine,
cos and sin are mixed: sin cos, cos sin
and the middle sign doesn't change: (-) → (-).
sin (A + B)
Example
Example
Solution
Set 15º = 45º - 30º.
You can also set
15º = 60º - 45º.
sin (45º - 30º)
= sin 45º cos 30º - cos 45º sin 30º
To find these trigonometric function values,
draw a 45-45-90 triangle
whose sides are 1, 1, √2,
and a 30-60-90 triangle
whose sides are 1, √3, 2.
sin 45º
Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (√2).
So sin 45º = 1/√2.
cos 30º
Cosine is CAH:
Cosine,
Adjacent side (√3),
Hypotenuse (2).
So cos 30º = √3/2.
Write -.
cos 45º
Cosine is CAH:
Cosine,
Adjacent side (1),
Hypotenuse (√2).
So cos 45º = 1/√2.
sin 30º
Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (2).
So sin 30º = 1/2.
So sin 45º cos 30º - cos 45º sin 30º
= [1/√2]⋅[√3/2] - [1/√2]⋅[1/2].
[1/√2]⋅[√3/2] - [1/√2]⋅[1/2]
= (√3 - 1)/2√2
To rationalize the denominator 2√2,
multiply [√2/√2].
(√3 - 1)√2 = √6 - √2
2√2⋅√2 = 2⋅2
2⋅2 = 4
So (√6 - √2)/4 is the answer.