sin (A + B)
How to find sin (A + B) by using its formula: formula, 2 examples, and their solutions.
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 1
Example
Solution
Set 105º = 60º + 45º.
sin (60º + 45º)
= sin 60º cos 45º + cos 60º sin 45º
To find these trigonometric function 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.
sin 60º
Sine is SOH:
Sine,
Opposite side (√3),
Hypotenuse (2).
So sin 60º = √3/2.
cos 45º
Cosine is CAH:
Cosine,
Adjacent side (1),
Hypotenuse (√2).
So cos 45º = 1/√2.
Write +.
cos 60º
Cosine is CAH:
Cosine,
Adjacent side (1),
Hypotenuse (2).
So cos 60º = 1/2.
sin 45º
Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (√2).
So sin 45º = 1/√2.
So sin 60º cos 45º + cos 60º sin 45º
= [√3/2]⋅[1/√2] + [1/2]⋅[1/√2].
[√3/2]⋅[1/√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.
Example 2
Example
Solution
To find the amplitude,
combine sin x and √3 cos x
by using sin (A + B) formula.
First write
y = 1 sin x + √3 cos x.
sin (x + B) = sin x cos B + cos x sin B
Your goal is
to change 1 to cos B
and to change √3 to sin B.
Cosine is CAH: related to the adjacent side.
Sine is SOH: related to the opposite side.
So draw a right triangle
whose adjacent side is 1
and whose opposite side is √3.
Find the hypotenuse
by using the Pythagorean theorem:
12 + (√3)2 = 22.
Then the hypotenuse is 2.
This is a right triangle
whose sides are 1, √3, and 2.
So this is a 30-60-90 triangle.
So the bottom angle is 60º.
Let's change
y = 1 sin x + √3 cos x.
First write the hypotenuse 2 and (.
1 sin x
= 2([1/2] sin x)
+√3 cos x
= 2( +[√3/2] cos x)
So 1 sin x + √3 cos x
= 2([1/2] sin x + [√3/2] cos x).
Next, change [1/2] and [√3/2]
to cosine and sine.
First write 2(.
To change [1/2],
see the right triangle.
Cosine is CAH:
Cosine,
Adjacent side (1),
Hypotenuse (2).
So 1/2 = cos 60º.
Write sin x.
Write +.
To change [√3/2],
see the right triangle.
Sine is SOH:
Sine,
Opposite side (√3),
Hypotenuse (2).
So √3/2 = sin 60º.
Write cos x).
So 2([1/2] sin x + [√3/2] cos x)
= 2(cos 60º sin x + sin 60º cos x).
cos 60º sin x + sin 60º cos x
= sin x cos 60º + cos x sin 60º
sin x cos 60º + cos x sin 60º
= sin (x + 60º)
See y = 2 sin (x + 60º).
The number in front of the sine is 2.
So the amplitude is
|2| = 2.
Sine: Graph
So 2 is the answer.