SME
Grade 12 Math · Precalculus
Sum, difference and double-angle formulas; trigonometric equations
Use this for “Find the exact value of sin 75° as sin(45° + 30°).”
Diagram
A + B = 45° + 30° = 75°: turn A, then B on
sin(75°)
sin(75°) = sin A cos B + cos A sin B
sin(75°)
=
√
2
2
×
√
3
2
+
√
2
2
×
1
2
sin(75°)
=
√
6
+
√
2
4
≈
0.9659
This is an example. Type your own values to start a new problem.
sin(
A
°
+
B
°)
=
S
C
The sum of the angles, A + B
worked out (not typed) · °
Formulas
C
=
A
+
B
75 = 45 + 30
S
=
sin(
A
°)
×
cos(
B
°)
+
cos(
A
°)
×
sin(
B
°)
√
6
+
√
2
4
=
sin(45°)
×
cos(30°)
+
cos(45°)
×
sin(30°)
Assumptions
•
sin(A + B) = sin A cos B + cos A sin B; it is not sin A + sin B.
•
Pick A and B with known exact values (30°, 45°, 60° and their relatives).
•
The same formula with minus signs gives sin(A − B) = sin A cos B − cos A sin B.
Step-by-step
WE KNOW
A = 45°, B = 30°
FIND
the sum of the angles, A + B (C), sin(A + B) (S)
Step 1 · Find the sum of the angles, A + B (C)
Use C = A + B
Add the two angles.
C
=
45
+
30
C
=
75°
Step 2 · Find sin(A + B): S
Use S = sin(A°) × cos(B°) + cos(A°) × sin(B°)
The
sum
formula:
sin
A
cos
B
plus
cos
A
sin
B
,
from
the
four
values
you
know.
S
=
sin(45°)
×
cos(30°)
+
cos(45°)
×
sin(30°)
sin
45°
=
√
2
2
,
cos
30°
=
√
3
2
,
cos
45°
=
√
2
2
,
sin
30°
=
1
2
√
2
2
×
√
3
2
+
√
2
2
×
1
2
=
√
6
4
+
√
2
4
=
√
6
+
√
2
4
S
=
√
6
+
√
2
4
≈
0.9659
Check
Put the numbers back in. Both sides should match.
75 = 45 + 30 ✓
√
6
+
√
2
4
=
sin(45°)
×
cos(30°)
+
cos(45°)
×
sin(30°)
✓
Topic check
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