SchoolDD.com provides concise explanations of trigonometric concepts like sine, cosine, and tangent functions. It explains how to use trigonometric functions to solve problems involving right triangles, with examples calculating values for angles like 30°, 60°, 37°, and 53° degrees. The site also summarizes trigonometric identity formulas and relationships between sine, cosine, and tangent for various angles.

1. SchoolDD.com
3
www.schoolDD.com

2. 3
F F F
F F F F
F FF F F F
F
3.1 ,
F F F F F F
F F F F F 45 F F F
45 450 F F
(Force) F F F F F
F F F F
F F F (N)
F
u=0 v v1 v2
F F
F v1 v2
(Mass) m F F F
(kg)
F
F F F
F F F
– F F www.schoolDD.com 1

3. (Weight) w F F m
F F g F g F F F F
F (N)
“ F ” m
W = mg
F
- (= )
- F FF F F F
F F F g F
- F F F F F F
F 2
F
F x-y
F
y
F
F sinθ F F x F cos θ
2
1 F F y F sin θ
θ x
F cosθ
F θ x F x F F F
F 1 x F F cosθ F y F
2 F F sinθ FF F FF cosθ F F
FF sinθ .... F F α y F F α F
F F F F .F . F F θ = 30°, α = 60° ( F F cos30° =
F sin 60° F sin 30° = F cos60° ?) F F !
F
- F F F F
F ( ) F F F
- F F F F
FF F
– F F www.schoolDD.com 2

4. F sin , cos , tan F
F sin , cos , tan F F F 30°, 60°, 37°, 53° 45°
F sin , cos , tan F F F F F F
F FF F F F F F F F
F F F F
F F
θ
2
c
F sinθ F F 1 2 F F
1
a F sinθ = a/c
θ
4 c F cosθ F F 3 4 F F
θ
F cosθ = b/c
b
3
a F tanθ F F 5 6 F F
5 F tanθ = a/b
θ
b 6
F F sin , cos , tan F F
F F F F F FF F F
F F F
F F F 30°, 60°, 37°,
53° 45°
60 53 45
2 5
1 3 1
30 37 45
4 1
F F F sin , cos , tan 30°, 60°, 37°,
53° 45° F F F F F
– F F www.schoolDD.com 3

5. F 1
F1 = 3 N F2 = 4 N m F
. F2 F1
m
+
F F= F 1 - F2
= 3-4
= -1 ( F F F F )
∴ F 1N F Ans
.
F2
60° F1
m
F2 x y F F1 F
y y
F2 sin60 F2 F F
ΣFy
60° x α x
F1 F2 cos60 ΣFX
x ΣFx = F1 + F2 cos60
= 3 + 4 (½ )
=5N
y ΣFy = F2 sin60
=4
=2 N
F F = Σ
=
= 6.08 N
– F F www.schoolDD.com 4

6. ∴ F 6.08 N Ans
Σ
F F tan α = Σ
= = 0.7
∴ α = tan-1 0.7 Ans
.
F1
F2
m
F F= F1 + F2
=3+4
=7N
∴ F 7N Ans
3.2
F (Sir Issac Newton) F F
F F F F
F F 3 F
F 1 F F
F F F F F
F 2 F F F F F F F
F F
F
F 3 Action = Reaction F F F
F
3 F
V=0 ,a=0 V ,a=0
=0
m m
F
F 1 F F F F
– F F www.schoolDD.com 5

7. F F
F F F
a
= ma
m
F F F
F 2 F F F F F F F F
F F F F F F F a F = ma
F21 F12
F12 = F21
F 3 Action = Reaction F F F F F12 F
F21 F F
F
- F F 3 F F F F
F F F F
- 3 F F F F F F
F 3 F F F F ( F F ..?) F F
F F F F F F F F F
F
F F
F F F
F F ?
F FF F F ?
F F ?
FF F F a F g
– F F www.schoolDD.com 6

8. F 2
F 20 kg F F F 24 m/s
8
.F . F F F F F FF F F
F F F ”
F
u=0( F ) v = 24 m/s
t=8 s
F=?
m m = 20 kg
m
F F F F F 2 = ma
F F F F
F =F F = ma --- (1)
F F Fm F Fa ∴ F a F
F a F v = u + at ( )
24 = 0 + a (8)
a = 3 m/s2
F a=3 (1) F
F = 20 (3)
∴ F = 60 N Ans
F
- F F F 2 F F F F
F F F F F F F
F F F F F F F
F F F F F F FF
– F F www.schoolDD.com 7

9. F 3
5 kg 100 m F
F
m = 5 kg
F=?
a =g
100 m
80
F = ma
F F = m(g)
F = 5(10)
∴ F = 50 N Ans
F
- F
F 4
200 N F F F 10 m/s2 F F 100 N
F F
F
a1 = 10 m/s2 a2 = ?
F1 = 200 N F2 = 100 N
m m
F F = ma
F F1 = ma1
∴m = = = 20 kg
F a2 F F = ma
F F2 = ma2
∴ a2 = = = 5 m/s2 Ans
– F F www.schoolDD.com 8

10. F F = ma
F = ma
m F F F a
=
F F =
∴ a2 = 5 m/s2 Ans
F 5
m1 F F 20 m/s2
m2 F F 5 m/s2 F m1/m2
F
a1 = 20 m/s2 a2 = 5 m/s2
=?
F F
m1 m2
F = ma
F F = m1a1 --- (1)
F = m2a2 --- (2)
(1) = (2) , m1a1 = m2a2
∴ = = = Ans
F F = ma
F = ma
F F F m
=
F F =
∴ = Ans
F
- F F F F F F
F F F
F F F ( )
F F F
– F F www.schoolDD.com 9

11. F 6
20 N F F F 10 F
50 m F F
u=0 m=?
F = 20 t = 10 F F
m m F ”
s = 50
= ma
F F = ma --- (1)
m FF F Fa a F
a F s = ut + ½ at2 ( )
50 = 0 + ½ a (102)
a = 1 m/s2
F a = 1 (1)
F = ma
20 = m(1)
m = 20 kg Ans
F 7
5 kg 6N 8N F
6N
5 kg
8N
= ma --- (1)
a Fm F F F
F 6N 8N F
F
10 N
= = = 10 N
8N F
6N tan = =
= 53
– F F www.schoolDD.com 10

12. F = 10 (1)
F a= = = 2 m/s2 F 53 6 N Ans
3.3 (Frictional force)
f F F
F F F F F
F F F
F F F F
F F
... .. ..
F
mg
f = N
f
N
f F F N
F 2
1. (Static friction) fs F
F F F F F F F
F F ( F )
2. F (Kinetic friction) fk F
.F . .. F F
F FF 0... fs (= sN) fs
F F fk (= kN) F F
F F F F ?
F
- s > F k
(F F F F .)
- F F F F
- ( sN) > F( kN)
- N
- W = mg F F
– F F www.schoolDD.com 11

13. F
?
F 8
F 10 kg F F
F 70 N F F F 40 N
. F F
. F F F F
F F F
v=0 F v
F = 70 N m = 10 kg F = 40 N
mg mg
fs = ? s =? fk = ? k=?
N N
. fs = ? , s = ?
fs
=
max x
F fs =
max
70 fs =
10(0) ( F ax = 0)
fs =
70 N Ans
s fs = sN --- (1)
y = may
N mg = may
N 10(10) = 10(0) ( F ay = 0)
N = 100 N
F N = 100 (1)
fs = sN
70 = s(100)
s = 0.7 Ans
– F F www.schoolDD.com 12

14. . fk = ? , k = ?
F F fk
= max x
F fk = max
40 fk = 10(0) ( F v ax = 0)
fk = 40 N Ans
F k fk = kN --- (1)
y = may
N mg = may
N 10(10) = 10(0) ( F ay = 0)
N = 100 N
F N = 100 (1)
fk = kN
40 = s(100)
k = 0.4 (< s ) Ans
F
- F mg N F
mg
mg mg
FF
N
N N
F FF mg
F FF F N F F
F F F F
F mg N F F ( F ) F
- F s F
– F F www.schoolDD.com 13

15. F 9
120 kg F F F F 20 m/s F
F 50 m F F
u = 20
v=0 F
t = 10 F F
m = 120 m f=? m
F ”
s = 50
F F u F
u = 20 m/s
F F( F ) F F
F 2 F F
F F F F
F F F 20 m/s F 1
= ma
F f = ma --- (1)
f Fm F Fa a F
a v2 = u2 + 2as ( )
0 = 202 + 2a (50)
a = -4 m/s2 (a F F u)
F a = -4 (1)
f = ma
f = 120 (-4)
f = -480 N ( F F )
F 480 N Ans
F 10
500 N 37 40 kg F
0.4 F F
500 N
37 N
F F ”
– F F www.schoolDD.com 14

16. y
F F .
500 sin37 500 a=?
F . F F F F
m = 40 37
mg 500 cos37 x F F F
f = 0.4 F F .
N
FF 500 N F F (x-y)
F
F a
x = max
F 500 cos37 - f = ma --- (1)
a Fm F Ff f F
f= N F F FN
N
= may y
F N + 500 sin37 - mg = may
N + 500(3/5) 40(10) = 0 ( F ay = 0)
N = 100 N
F F f = N = 0.4(100) = 40 N
F f = 40 (1)
500 cos37 - f = ma
500(4/5) - 40 = 40a
a = 9 m/s2 Ans
F
- F = ma F 2 ( F)
F F m F F a F FF
F F F F
F F
a
N = ma
T F F F F = ma
m
F T f = ma
mg f
– F F www.schoolDD.com 15

17. F 11
F F F
F s = 1.0 k = 0.80 F F F
F F F F F F F
F F F F FF
F F F
“ F F F
F F F F 3
w = mg , f
,
N 2
mg sinθ mg cosθ ”
90-θ
θ
N
F 180 F f
mg sinθ θ
mg cosθ
θ
W = mg
F F F F
F
= ma
F F F = ma
F mg sin - fs = ma F a = 0 ( F )
mg sin = fs --- (1)
fs fs = s N F s F F N
N
= ma
F N - mg cos = ma
N - mg cos = 0 ( F a = 0)
N = mg cos
– F F www.schoolDD.com 16

18. F N F fs = s N = 1.0 mg cos = mg cos
F fs (1)
mg sin = mg cos
θ
θ
= 1.0
tan = 1.0
= 45° Ans
F F F F F
F
= ma
F F F = ma
F mg sin fk = ma
mg sin - k N = ma
mg sin - k (mg cos ) = ma (N = mg cos )
10 ( ) - 0.80 (10)( ) = a ( F m F )
a = = 1.414 m/s2 Ans
F
- F F F F F m F
F F F
F F F F F F F FF F F F F
F F
F 12
90 N 10 kg F F 20 kg F
a=? . F
. F
90 N 20 kg . F 90 N F 20 kg F
10 kg
F F
F
. F a=?
= ma
– F F www.schoolDD.com 17

19. F F F = ma
m = 10 + 20 = 30 kg ( F F )
F 90 = 30 (a)
a = 3 m/s2 Ans
. F R
F F F
10 kg
a = 3 m/s2 F R F - F
F F
90 N R
10kg F F 3 ”
= ma
F F F = ma
90 - R = 10 (3)
R = 60 N Ans
20 kg
a = 3 m/s2
R 20 kg
= ma
F F F = ma
R = 20 (3)
R = 60 N Ans ( F 10 kg)
F F F F F F
.
a=?
20 kg 90 N
10 kg
F a=?
– F F www.schoolDD.com 18

20. = ma
F F F = ma
m = 10 + 20 = 30 kg
F 90 = 30 (a)
a = 3 m/s2 Ans ( F F FF F )
20 kg
a = 3 m/s2
R 90 N
20 kg
= ma
F F F = ma
90 - R = 20 (3)
R = 30 N Ans ( F F F F .)
10 kg
a = 3 m/s2
R
10kg
= ma
F F F = ma
R = 10 (3)
R = 30 N Ans
F
- F mg
N F F F F F
F F F F F F F F F F
– F F www.schoolDD.com 19

21. F 13
F 120 N 5 kg 10 kg 15 kg F F
F T1 T2 F = 0.10
15 kg T1 10 kg T2 F
5 kg
F a
F ”
a=?
N1 N2 N3
m1 T1 T2 F
m2 m3
f1 = 0.10 f2 f3
m1g m2g m3g
= ma
F F F = ma
F F - f1 - f2 - f3 = ma
F - N1 - N2 - N3 = (m1+ m2 + m3)a
120 - (N1 + N2 + N3 ) = (15 + 10 + 5)a
120 - 0.1(15(10) + 10(10) + 5(10)) = 30 a ( F N = mg)
a = 3 m/s2 Ans
F T1
15 kg
T1 F F F 10 kg 15 kg 15 kg F m1
a = 3 m/s2
N1
T1 = ?
m1
f1
m1g
= ma
– F F www.schoolDD.com 20

22. F F F = ma
F T1 - f1 = m1a
T1 - N1 = m1a
T1 - m1g = m1a
T1 = m1(a + g)
T1 = 15(3 + 0.1(10))
T1 = 60 N Ans
F T2
5 kg
N3 a = 3 m/s2
T2 = ? F = 120 N
m3
m3g f3
= ma
F F F = ma
F F T2 f3 = m3a
F T2 N3 = m3a
F T2 (m3g) = m3a
120 T2 0.1(5x10) = 5(3)
T2 = 100 N Ans
F
- F F F F
F F F F F F
F F F F F F F F
F F F F
T F
m1 m2
T
T
T
m m T T F
m1 m2
mg mg
– F F www.schoolDD.com 21

23. F 14
80 N 2 kg 3 kg F F F F
F
F = 80 N
2 kg
T a
3 kg
F a
F F T F F
F = 80 N
m1
m1g a=?
m2
m2g
= ma
F F F = ma
F F m1g m2g = (m1 + m2)a
80 2(10) 3(10) = (2 + 3)a
a = 6 m/s2 Ans
F T
3 kg
T=?
m2 a = 6 m/s2
m2g
– F F www.schoolDD.com 22

24. = ma
F F F = ma
F T m2g = m2a
T 3(10) = 3(6)
T = 48 N Ans
F 15
2 kg 3 kg F F F F F
F F
T T
2 3
F a
3 kg F F 2 kg F
F a F F F
F
”
a=? m1 m2
a=?
m1g m2g
= ma
F F F = ma
F m2g m1g = (m1 + m2)a
3(10) 2(10) = (2 + 3)a
a = 2 m/s2 Ans
– F F www.schoolDD.com 23

25. F
2 kg
T=?
2
a = 2 m/s m1
m1g
= ma
F F F = ma
F T m1g = m1a
T 2(10) = 2(2)
T = 24 N Ans
F 16
F F F F F F
T1
T1
1010 kg
kg 25 kg
T2
37°
T2
15 kg
F a
a
N F F
102kg
m m1
a F a
m2g sin37
F F F F
37° m2g m2g cos37 m1g
a m3
F
m3g
F m1
= ma
– F F www.schoolDD.com 24

26. F F F = ma
F m1g m2g sin37 m3g = (m1 + m2 + m3)a
25(10) 10 (10)(3/5) 15(10) = (25 + 10 + 15)a
a = 0.8 m/s2 Ans
F a F F a F
F T1
m1
T1 = ?
m1 a = 0.8 m/s2
m1g
= ma
F F F = ma
F m1g T1 = m1a
25(10) T1 = 25(0.8)
T1 = 230 N Ans
F T2
m3
T2 = ?
m3 a = 0.8 m/s2
m3g
= ma
F F F = ma
F T2 m3g = m3a
T2 15(10) = 15(0.8)
T2 = 162 N Ans
– F F www.schoolDD.com 25

27. F
- F F F F F F
T2 = 2T1 T2 = T1/2
T2 = T1/2
T1
T1
m1 T1
1
m2 m1
1 1
3.4
F F F F F F
mg
T T
m
m
T T
T N
m1 m2
m
m1g m2g
mg
mg
1 2 3
1 2 F F F F T
3 F F F F F N
F F F F F F ?
F F F F ?
F 17
F 1 kg F F F F F F F
. F
. F
. F
. F F 0.5 m/s2
. F F 0.5 m/s2
m = 1 kg
m
– F F www.schoolDD.com 26

28. F F F F
. F
F F
T
m a=0
mg
= ma
F F F = ma
F T mg = ma
T 1(10) = 1(0)
T = 10 N ( F ) Ans
. F
F F
T
m v ,a=0
mg
= ma
F F F = ma
F T mg = ma
T 1(10) = 1(0)
T = 10 N ( F ) Ans
. F
F F
T
m v ,a=0
mg
– F F www.schoolDD.com 27

29. = ma
F F F = ma
F mg T = ma
1(10) T = 1(0)
T = 10 N ( F ) Ans
. F F 0.5 m/s2
F F
T
m a = 0.5 m/s2
mg
= ma
F F F = ma
F T mg = ma
T 1(10) = 1(0.5)
T = 10.5 N ( F ) Ans
. F F 0.5 m/s2
F F
T
m a = 0.5 m/s2
mg
= ma
F F F = ma
F mg T = ma
1(10) T = 1(0.5)
T = 9.5 N ( F F ) Ans
– F F www.schoolDD.com 28

30. F 18
5 kg 15 kg F F F F
15 5
F F F F F F F
F a F
T T
T T
a m1 m2 a
m1g m2g
mg
F 15 kg F 5 kg
F a
= ma
F F F = ma
F m1g m2g = (m1 + m2) a
15(10) 5(10) = (15 + 5)a
a = 5 m/s2
T
m2
T=?
m2 a = 5 m/s2
m2g
= ma
F F F = ma
F T m2g = m2a
T 5(10) = 5(5)
T = 75 N Ans
– F F www.schoolDD.com 29

31. F 19
F 100 kg F F F F
F F F F F F
F F F F F 5 m/s2 F
F F F F F F F F F
F
F F FF F F F
a = 5 m/s2 N F
mg F F F 3
”
N
F F a = 5 m/s2 F
( )
= ma
F F F = ma
F mg N = ma
100(10) N = 100(5)
N = 500 N
F F = 50 F Ans
F F F !!”
F 20
F F 20 F F F F F F
. F F
. F F F 1 m/s2
. F F F 1 m/s2
. F F 1 m/s
. F F F 1 m/s2
. F F F 1 m/s2
. F F 1 m/s
. F
– F F www.schoolDD.com 30

32. . a=0 F F
F ”
= ma
N - mg = 0
a=0
mg N = mg
N = 100 (10) = 1000 N
N F F = 100 Ans
. F F F a = 1 m/s2
= ma
N - mg = ma F F F F
2
a = 1 m/s F
N = m (a + g)
mg F 2 F ”
N = 100 (1 + 10)
N
N = 1100 N
F F = 110 Ans
. F F F a = -1 m/s2
= ma
N - mg = ma F F F a - F
a = -1 m/s2 F
mg N = m (a + g)
F F ”
N = 100 (-1 + 10)
N N = 900 N
F F = 90 Ans
. F F v = 1 m/s a=0
= ma
v
N - mg = 0 F F F
mg a=0
N = mg F F
”
N N = 100 (10)
N = 1000 N
F F = 100 Ans
. F F F a = 1 m/s2
= ma
2
a = 1 m/s mg - N = ma
mg
N = m (g - a)
N = 100 (10 - 1)
N
– F F www.schoolDD.com 31

33. N = 900 N
F F = 90 Ans
. F F F a = -1 m/s2
= ma
a = -1 m/s2 mg - N = ma
mg N = m (g - a)
N = 100 (10 - (-1))
N
N = 1100 N
F F = 110 Ans
. F F v = 1 m/s a=0
= ma
mg - N = 0
v
mg a=0 N = mg
N = 100 (10)
N N = 1000 N
F F = 100 Ans
. F
a=g F !! ! F F F
a=g = ma F F
mg mg - N = mg F F a = g”
N = 0 N
N F F= 0 Ans
F 21
50 kg F F F F 0.5 m/s2
F F F F 40 kg F
– F F www.schoolDD.com 32

34. ( ) F
T=?
a = 0.5
mg
N = 400 N
= ma
F F F = ma
F mg N T = ma
50(10) 400 T = 50(0.5)
T = 75 N Ans
F 22
F 40 kg F 30
F F
30
( F ) F
a
mg
ax = a cos30 ay mg
mg sin30 30°
N N=?
a ay = a sin30
30
FF F F F N F FF
F F F F N
– F F www.schoolDD.com 33

35. F mg sin30
mg
= ma
F F F = ma
F mg sin30 = ma
a = g sin30
= 10(1/2)
a = 5 m/s2
ay = a sin30 = (5x1/2) = 2.5 m/s2
= ma
F F F = ma
F mg N = may
40(10) N = 40(2.5)
N = 300 N
F F = 30 Ans
F 23
70 kg F F
. F F 1 m/s2
. F F 1 m/s2
F
( ) F
T mg”
. F F a = 1 m/s2
T
a = 1 m/s2
mg
= ma
– F F www.schoolDD.com 34

36. F F F = ma
F T mg = ma
T 70(10) = 70(1)
T = 770 N Ans
. F F a = 1 m/s2
T
a = 1 m/s2
mg
= ma
F F F = ma
F mg T = ma
70(10) T = 70(1)
T = 630 N Ans
F F F F F ?”
F 24
F F 8 kg F F F 10 kg F
F F F F F F
F F F F F F F
F F F F a1 = 0
F
T
a1 = 0
m1g
= ma
– F F www.schoolDD.com 35

37. F T = m1g a=0
T = 10(10)
T = 100 N
T
a2 = ?
m2g
= ma
F F F = ma
F T m2g = m2a2
100 8(10) = 8 a2
a2 = 2.5
F F F F 2.5 m/s2 Ans
F 25
F 2 F F F 0.25
F F
FF F F F
F F 3 F
4kg ”
F = 100 N
6 kg
F F F
a N1 a
m1g f1
T T m2g
F = 100 N
f1 f2
N1 N2
FF N1 F
F F F F N2
F F F F F
– F F www.schoolDD.com 36

38. F m1 = 4
= ma
F F F = ma
F T f1 = m1a
T N1 = m1a
T 0.25(4x10) = 4a
T = 4a + 10 ---- (1)
F F m2 = 6
= ma
F F F = ma
F F T f1 f2 = m2a
F T (1) f = µN
F (4a + 10) N1 N2 = m2a
N1 = m1g , N2 = N1 + m2g F
F (4a + 10) m1g (N1 + m2g) = m2a
100 (4a + 10) 0.25(4x10) 0.25(4x10 + 6x10) = 6a
a = 5.5 m/s2 Ans
F a (1)
T = 4a + 10
T = 4(5.5) + 10
T = 32 N Ans
F F F
= ma
F F F = ma
F - f2 - f1 - f1 = (m1 + m2)a
100 - 0.25(10x10) - 0.25(4x10) - 0.25(4x10) = 10a
a = 5.5 m/s2 Ans
– F F www.schoolDD.com 37

39. F 26
F F F F F F F F
F F F 0.50 ( F )
a
FF F F F FF F
F
F F F
F
F
f a=?
N
mg
= ma
F F F = ma
F N = ma --- (1)
F
F f = mg
N = mg
N = mg/
F N (1)
N = ma --- (1)
mg/ = ma
a = g/ F F
a = 10/ 0.5
a = 20 m/s2 Ans
– F F www.schoolDD.com 38

40. F 27
F F F F F F
2
F F 10 m/s
a = 10 m/s2
F
Tcos F
2
a = 10 m/s
T ”
Tsin
mg
= ma
F F F = ma
F Tsin = ma --- (1)
F
F Tcos = mg --- (2)
(1)/ (2) , Tsin / Tcos = ma/mg
tan = a/g
tan = 10/10
tan = 1
= 45 Ans
F
- FF F F F ! F F
F F F F F F
F F F F F F
F F F F
– F F www.schoolDD.com 39