Trigonometric Ratio of Associated Angles

Question 1

sin²θ + cos²θ = ?

Answer 1

sin²θ + cos²θ = 1

Question 2

sec²θ - tan²θ = ?

Answer 2

sec²θ - tan²θ = 1

Question 3

cosec²θ - cot²θ = ?

Answer 3

cosec²θ - cot²θ = 1

Question 4

sin(A+B) = ?

Answer 4

sin(A+B) = sinAcosB + cosAsinB

Question 5

sin(A-B) = ?

Answer 5

sin(A-B) = sinAcosB - cosAsinB

Question 6

cos(A+B) = ?

Answer 6

cos(A+B) = cosAcosB - sinAsinB

Question 7

cos(A-B) = ?

Answer 7

cos(A-B) = cosAcosB + sinAsinB

Question 8

tan(A+B) = ?

Answer 8

tan(A+B) = (tanA + tanB)/(1 - tanAtanB)

Question 9

tan(A-B) = ?

Answer 9

tan(A-B) = (tanA - tanB)/(1 + tanAtanB)

Question 10

cot(A+B) = ?

Answer 10

cot(A+B) = (cotAcotB - 1)/(cotB + cotA)

Question 11

cot(A-B) = ?

Answer 11

cot(A-B) = (cotAcotB + 1)/(cotB - cotA)

Question 12

tan(A+B+C) = ?

Answer 12

tan(A+B+C) = (tanA + tanB + tanC - tanAtanBtanC)/(1 - tanAtanB - tanBtanC - tanCtanA)

Question 13

sin(A+B)sin(A-B) = ?

Answer 13

sin(A+B)sin(A-B) = sin²A - sin²B = cos²B - cos²A

Question 14

cos(A+B)cos(A-B) = ?

Answer 14

cos(A+B)cos(A-B) = cos²A - sin²B = cos²B - sin²A

Question 15

sin(A+B) + sin(A-B) = ?

Answer 15

sin(A+B) + sin(A-B) = 2sinAcosB

Question 16

sin(A+B) - sin(A-B) = ?

Answer 16

sin(A+B) - sin(A-B) = 2cosAsinB

Question 17

cos(A+B) + cos(A-B) = ?

Answer 17

cos(A+B) + cos(A-B) = 2cosAcosB

Question 18

cos(A-B) - cos(A+B) = ?

Answer 18

cos(A-B) - cos(A+B) = 2sinAsinB

Question 19

sinC + sinD = ?

Answer 19

sinC + sinD = 2sin((C+D)/2)cos((C-D)/2)

Question 20

sinC - sinD = ?

Answer 20

sinC - sinD = 2cos((C+D)/2)sin((C-D)/2)

Question 21

cosC + cosD = ?

Answer 21

cosC + cosD = 2cos((C+D)/2)cos((C-D)/2)

Question 22

cosC - cosD = ?

Answer 22

cosC - cosD = 2sin((C+D)/2)sin((D-C)/2) = -2sin((C+D)/2)sin((C-D)/2)

Question 23

sin2A = ?

Answer 23

sin2A = 2sinAcosA = (2tanA)/(1 + tan²A)

Question 24

cos2A = ?

Answer 24

cos2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A = (1 - tan²A)/(1 + tan²A)

Question 25

sin3A = ?

Answer 25

sin3A = 3sinA - 4sin³A

Question 26

cos3A = ?

Answer 26

cos3A = 4cos³A - 3cosA

Question 27

tan2A = ?

Answer 27

tan2A = (2tanA)/(1 - tan²A)

Question 28

cot2A = ?

Answer 28

cot2A = (cot²A - 1)/(2cotA)

Question 29

tan3A = ?

Answer 29

tan3A = (3tanA - tan³A)/(1 - 3tan²A)

Question 30

cot3A = ?

Answer 30

cot3A = (cot³A - 3cotA)/(3cot²A - 1)

Question 31

2cos²A = ?

Answer 31

2cos²A = 1 + cos2A

Question 32

2sin²A = ?

Answer 32

2sin²A = 1 - cos2A

Question 33

tan²A = ?

Answer 33

tan²A = (1 - cos2A)/(1 + cos2A)

Question 34

4sin³A = ?

Answer 34

4sin³A = 3sinA - sin3A

Question 35

4cos³A = ?

Answer 35

4cos³A = 3cosA + cos3A

Question 36

tan(θ/2) = ?

Answer 36

tan(θ/2) = √(1 - cosθ) / √(1 + cosθ) = (sinθ)/(1 + cosθ) = (1 - cosθ)/(sinθ)

Question 37

cot(θ/2) = ?

Answer 37

cot(θ/2) = √(1 + cosθ)/√(1 - cosθ) = (1 + cosθ)/(sinθ) = (sinθ)/(1 - cosθ)

Question 38

Δ = ?

Answer 38

Δ = (1/2)absin(C) = (1/2)bcsin(A) = (1/2)casin(B)

Question 39

sine Rule = ?

Answer 39

a/sinA = b/sinB = c/sinC = 2R

Question 40

cosine Rule = ?

Answer 40

cosA = (b² + c² - a²)/(2bc); cosB = (c² + a² - b²)/(2ca); cosC = (a² + b² - c²)/(2ab);