Trig Identities

Question 1

Pythagorean identity sin^2(x)

Answer 1

sin^2(x) = 1 - cos^2(x)

Question 2

Pythagorean identity cos^2(x)

Answer 2

cos^2(x) = 1 - sin^2(x)

Question 3

Pythagorean identity tan^2(x)

Answer 3

tan^2(x) = sec^2(x) - 1

Question 4

Pythagorean identity cot^2(x)

Answer 4

cot^2(x) = csc^2(x) - 1

Question 5

Reciprocal identity for sin(x)

Answer 5

sin(x) = 1/csc(x)

Question 6

Reciprocal identity for cos(x)

Answer 6

cos(x) = 1/sec(x)

Question 7

Reciprocal identity for tan(x)

Answer 7

tan(x) = 1/cot(x)

Question 8

Reciprocal identity for csc(x)

Answer 8

csc(x) = 1/sin(x)

Question 9

Reciprocal identity for sec(x)

Answer 9

sec(x) = 1/cos(x)

Question 10

Reciprocal identity for cot(x)

Answer 10

cot(x) = 1/tan(x)

Question 11

Quotient identity for tan(x)

Answer 11

tan(x) = sin(x)/cos(x)

Question 12

Quotient identity for cot(x)

Answer 12

cot(x) = cos(x)/sin(x)

Question 13

Sum identity for sin(a+b)

Answer 13

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

Question 14

Sum identity for sin(a-b)

Answer 14

sin(a-b) = sin(a)cos(b) - cos(a)sin(b)

Question 15

Sum identity for cos(a+b)

Answer 15

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

Question 16

Sum identity for cos(a-b)

Answer 16

cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

Question 17

Sum identity for tan(a+b)

Answer 17

tan(a+b) = [tan(a) + tan(b)]/[1 - tan(a)tan(b)]

Question 18

Sum identity for tan(a-b)

Answer 18

tan(a-b) = [tan(a) - tan(b)]/[1 + tan(a)tan(b)]

Question 19

Double angle identity for sin(2x)

Answer 19

sin(2x) = 2sin(x)cos(x)

Question 20

Double angle identity for cos(2x) (first form)

Answer 20

cos(2x) = cos²(x) - sin²(x)

Question 21

Double angle identity for cos(2x) (second form)

Answer 21

cos(2x) = 2cos²(x) - 1

Question 22

Double angle identity for cos(2x) (third form)

Answer 22

cos(2x) = 1 - 2sin²(x)

Question 23

Double angle identity for tan(2x)

Answer 23

tan(2x) = 2tan(x)/[1 - tan²(x)]

Question 24

Half angle identity for sin(x/2) (with positive radical)

Answer 24

sin(x/2) = ±√[(1 - cos(x))/2]

Question 25

Half angle identity for cos(x/2) (with positive radical)

Answer 25

cos(x/2) = ±√[(1 + cos(x))/2]

Question 26

Half angle identity for tan(x/2)

Answer 26

tan(x/2) = [1 - cos(x)]/sin(x) = sin(x)/[1 + cos(x)]

Question 27

Product-to-sum identity for sin(a)sin(b)

Answer 27

sin(a)sin(b) = [cos(a-b) - cos(a+b)]/2

Question 28

Product-to-sum identity for cos(a)cos(b)

Answer 28

cos(a)cos(b) = [cos(a-b) + cos(a+b)]/2

Question 29

Product-to-sum identity for sin(a)cos(b)

Answer 29

sin(a)cos(b) = [sin(a+b) + sin(a-b)]/2

Question 30

Sum-to-product identity for sin(a) + sin(b)

Answer 30

sin(a) + sin(b) = 2sin[(a+b)/2]cos[(a-b)/2]

Question 31

Sum-to-product identity for sin(a) - sin(b)

Answer 31

sin(a) - sin(b) = 2cos[(a+b)/2]sin[(a-b)/2]

Question 32

Sum-to-product identity for cos(a) + cos(b)

Answer 32

cos(a) + cos(b) = 2cos[(a+b)/2]cos[(a-b)/2]

Question 33

Sum-to-product identity for cos(a) - cos(b)

Answer 33

cos(a) - cos(b) = -2sin[(a+b)/2]sin[(a-b)/2]

Question 34

Power reducing formula for sin²(x)

Answer 34

sin²(x) = [1 - cos(2x)]/2

Question 35

Power reducing formula for cos²(x)

Answer 35

cos²(x) = [1 + cos(2x)]/2

Question 36

Power reducing formula for tan²(x)

Answer 36

tan²(x) = [1 - cos(2x)]/[1 + cos(2x)]

Question 37

Even/odd property of sin(-x)

Answer 37

sin(-x) = -sin(x)

Question 38

Even/odd property of cos(-x)

Answer 38

cos(-x) = cos(x)

Question 39

Even/odd property of tan(-x)

Answer 39

tan(-x) = -tan(x)

Question 40

Law of sines

Answer 40

sin(A)/a = sin(B)/b = sin(C)/c

Question 41

Law of cosines (first form)

Answer 41

c² = a² + b² - 2ab·cos(C)

Question 42

Law of cosines (angle form)

Answer 42

cos(C) = (a² + b² - c²)/(2ab)