SME
Grade 12 Math · Problem type
The difference formula for cosine
Use this for “Find the exact value of cos 15° as cos(45° − 30°).”
Diagram
A − B = 45° − 30° = 15°: turn A, then B back
cos(15°)
cos(15°) = cos A cos B + sin A sin B
cos(15°)
=
√
2
2
×
√
3
2
+
√
2
2
×
1
2
cos(15°)
=
√
6
+
√
2
4
≈
0.9659
This is an example. Type your own values to start a new problem.
cos(
A
°
−
B
°)
=
K
C
The difference of the angles, A − B
worked out (not typed) · °
Formulas
C
=
A
−
B
15 = 45 − 30
K
=
cos(
A
°)
×
cos(
B
°)
+
sin(
A
°)
×
sin(
B
°)
√
6
+
√
2
4
=
cos(45°)
×
cos(30°)
+
sin(45°)
×
sin(30°)
Assumptions
•
cos(A − B) = cos A cos B + sin A sin B: the sign in the middle flips.
•
Write the angle as a difference of angles with known exact values.
•
cos(A + B) = cos A cos B − sin A sin B is the same formula with B made negative.
Step-by-step
WE KNOW
A = 45°, B = 30°
FIND
the difference of the angles, A − B (C), cos(A − B) (K)
Step 1 · Find the difference of the angles, A − B (C)
Use C = A − B
Take the second angle from the first.
C
=
45
−
30
C
=
15°
Step 2 · Find cos(A − B): K
Use K = cos(A°) × cos(B°) + sin(A°) × sin(B°)
The
difference
formula:
cos
A
cos
B
plus
sin
A
sin
B
.
K
=
cos(45°)
×
cos(30°)
+
sin(45°)
×
sin(30°)
cos
45°
=
√
2
2
,
cos
30°
=
√
3
2
,
sin
45°
=
√
2
2
,
sin
30°
=
1
2
√
2
2
×
√
3
2
+
√
2
2
×
1
2
=
√
6
4
+
√
2
4
=
√
6
+
√
2
4
K
=
√
6
+
√
2
4
≈
0.9659
Check
Put the numbers back in. Both sides should match.
15 = 45 − 30 ✓
√
6
+
√
2
4
=
cos(45°)
×
cos(30°)
+
sin(45°)
×
sin(30°)
✓
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