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Pre-Calculus: Angle Operations Identities

Eyles’ Pre-calculus ANGLE OPERATIONS IDENTITIES
1. Assume that cos(a – ß) = cosa cosß + sina sinß
BONUS:
Prove the following identities. ( HINT S )
2. cos(90? – ß) = sinß apply#1
3. sin(90? – ß) = cosß ß = 90? – (90? – ß)
4. cos(-ß) = cosß – ß = 0? – ß
5. sin(-ß) = -sinß apply#2
6. cos(a + ß) = cosa cosß – sina sinß apply#1 then #5
7. sin(a + ß) = sina cosß + cosa sinß apply#2 then #1 then #3
8. sin(a – ß) = sina cosß – cosa sinß apply#7 then #2 then #3
9. sin 2ß = 2sinß cosß apply#7
10. cos 2ß = cos2ß – sin2ß apply#6
11. cos 2ß = 2cos2ß – 1 apply#10
12. cos 2ß = 1 – 2sin2ß apply#10
13. sin
a
2
= ±
s
1 – cosa
2
apply#12
14. cos
a
2
= ±
s
1 + cosa
2
apply#11
15. Derive Angle Operations Identity # 1 above using the distance formula.
Eyles’ Pre-calculus ANGLE OPERATIONS PRACTICE PROBLEMS
Apply the Angle Operations Identities on the previous page to Find the EXACT values.
16. a) cos(15?
) = b) sin(105?
) c) sin(-75?
)
17. a) cos(
11p
12
) = b) sin(
-13p
12
) c) sin(
-p
8
)
18. a) cos(35?
)cos(5?
) + sin(35?
)sin(5?
) b) cos(95?
)cos(55?
) – sin(95?
)sin(55?
)
19. a) cos(143?
)cos(83?
) + sin(143?
)sin(83?
) b) cos(177?
)cos(33?
) – sin(177?
)sin(33?
)
20. a) sin(35?
)cos(10?
) + cos(35?
)sin(10?
) b) sin(950?
)cos(650?
) – cos(950?
)sin(650?
)
21. a) sin(143?
)cos(37?
) + cos(143?
)sin(37?
) b) sin(177?
)cos(27?
) – cos(177?
)sin(27?
)
22. a) cos(22.5
?
) = b) sin(-67.5
?
) c) sin(112.5
?
)
23. a) cos(
-p
12
) = b) sin(
-p
8
) c) sin(
-3p
8
)
If A B are in quadrant I and sinA =
2
5
and sinB =
10
11
find
24. a) cos(A + B) b) cos(A – B)
25. a) sin(A + B) b) sin(A – B)
26. a) cos(2A) b) sin(2A)
27. a) cos(
A
2
) b) sin(
A
2
)
2

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