Trigonometry

Question 1

1 − 𝑠𝑖𝑛^2 𝜃 =

Answer 1

𝑐𝑜𝑠^2 𝜃

Question 2

1 + 𝑡𝑎𝑛^2 𝜃 =

Answer 2

𝑠𝑒𝑐^2 𝜃

Question 3

1 + 𝑐𝑜𝑡^2 𝜃 =

Answer 3

𝑐𝑜𝑠𝑒𝑐^2 𝜃

Question 4

𝑠𝑖𝑛^4 𝜃 + 𝑐𝑜𝑠^4 𝜃 =

Answer 4

1 − 2 𝑠𝑖𝑛^2 𝜃 𝑐𝑜𝑠^2 𝜃

Question 5

𝑠𝑖𝑛^6 𝜃 + 𝑐𝑜𝑠^6 𝜃 =

Answer 5

1 − 3 𝑠𝑖𝑛^2 𝜃 𝑐𝑜𝑠^2 𝜃

Question 6

If 𝑥 = 𝑎𝑠𝑖𝑛 𝜃 + 𝑏 𝑐𝑜𝑠 𝜃 & 𝑦 = a cos 𝜃 − 𝑏 sin 𝜃 OR 𝑥 = 𝑎 𝑠𝑖𝑛 𝜃 + 𝑏 𝑐𝑜𝑠 𝜃 & 𝑦 = 𝑏 𝑠𝑖𝑛 𝜃 − 𝑎 𝑐𝑜𝑠 𝜃
then

Answer 6

𝑥^2 + 𝑦^2 = 𝑎^2 + 𝑏^2

Approach: square & add the equations

Question 7

𝑠𝑖𝑛(𝐴 + 𝐵) =

Answer 7

𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 + 𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 B

Question 8

𝑠𝑖𝑛(𝐴 − 𝐵) =

Answer 8

𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 − 𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 B

Question 9

𝑐𝑜𝑠(𝐴 + 𝐵) =

Answer 9

𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠 𝐵 − 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 B

Question 10

𝑐𝑜𝑠(𝐴 − 𝐵) =

Answer 10

𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠 𝐵 + 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 B

Question 11

𝑡𝑎𝑛(𝐴 + 𝐵) =

Answer 11

𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵 / 1 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 B

Question 12

𝑡𝑎𝑛(𝐴 − 𝐵) =

Answer 12

𝑡𝑎𝑛 𝐴 − 𝑡𝑎𝑛 𝐵 / 1 + 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵

Question 13

𝑐𝑜𝑡(𝐴 + 𝐵) =

Answer 13

𝑐𝑜𝑡 𝐴 𝑐𝑜𝑡 𝐵 − 1 / 𝑐𝑜𝑡 𝐴 + 𝑐𝑜𝑡 𝐵

Question 14

𝑡𝑎𝑛 (𝜋/4+ 𝜃) =

Answer 14

= 1 + 𝑡𝑎𝑛 𝜃 / 1 − 𝑡𝑎𝑛 𝜃

= cos 𝜃 + sin 𝜃 / cos 𝜃 − sin 𝜃

Question 15

𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃 =

Answer 15

√2 𝑠𝑖𝑛 (𝜃 +𝜋/4)

Question 16

𝑠𝑖𝑛(𝐴 + 𝐵) 𝑠𝑖𝑛(𝐴 − 𝐵) =

Answer 16

𝑠𝑖𝑛^2 𝐴 − 𝑠𝑖𝑛^2 𝐵

Question 17

𝑐𝑜𝑠(𝐴 + 𝐵) 𝑐𝑜𝑠(𝐴 − 𝐵) =

Answer 17

𝑐𝑜𝑠2 𝐴 − 𝑠𝑖𝑛2 B

Question 18

(1 + 𝑡𝑎𝑛 𝜃)(1 + 𝑡𝑎𝑛(45° − 𝜃)) =

Answer 18

2

Question 19

2 𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 =

Answer 19

𝑠𝑖𝑛(𝐴 + 𝐵) + 𝑠𝑖𝑛(𝐴 − 𝐵)

Question 20

2 𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 𝐵 =

Answer 20

𝑠𝑖𝑛(𝐴 + 𝐵) − 𝑠𝑖𝑛(𝐴 − 𝐵)

Question 21

2 𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠 𝐵 =

Answer 21

𝑐𝑜𝑠(𝐴 + 𝐵) + 𝑐𝑜𝑠(𝐴 − 𝐵)

Question 22

2 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵 =

Answer 22

𝑐𝑜𝑠(𝐴 − 𝐵) − 𝑐𝑜𝑠(𝐴 + 𝐵)

Question 23

𝑠𝑖𝑛 𝐶 + 𝑠𝑖𝑛 𝐷 =

Answer 23

2 𝑠𝑖𝑛 𝐶+𝐷 /2 𝑐𝑜𝑠 𝐶−𝐷 /2

Question 24

𝑠𝑖𝑛 𝐶 − 𝑠𝑖𝑛 𝐷 =

Answer 24

2 𝑐𝑜𝑠 𝐶 + 𝐷 / 2 𝑠𝑖𝑛 𝐶 − 𝐷 /2

Question 25

𝑐𝑜𝑠 𝐶 + 𝑐𝑜𝑠 𝐷 =

Answer 25

2 𝑐𝑜𝑠 𝐶 + 𝐷 /2 𝑐𝑜𝑠 𝐶 − 𝐷 /2

Question 26

𝑐𝑜𝑠 𝐶 − 𝑐𝑜𝑠 𝐷 =

Answer 26

−2 𝑠𝑖𝑛 𝐶 + 𝐷 /2 𝑠𝑖𝑛 𝐶 − 𝐷 /2

Question 27

𝑠𝑖𝑛 2𝜃 =

Answer 27

2 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃

Question 28

𝑐𝑜𝑠 2𝜃 =

Answer 28

=𝑐𝑜𝑠^2 𝜃 − 𝑠𝑖𝑛^2 𝜃

= 2 𝑐𝑜𝑠^2 𝜃 − 1

= 1 − 2 𝑠𝑖𝑛^2 𝜃

Question 29

1 − 𝑐𝑜𝑠 2𝜃 =

Answer 29

2 𝑠𝑖𝑛^2 𝜃

Question 30

1 + 𝑐𝑜𝑠 2𝜃 =

Answer 30

2 𝑐𝑜𝑠^2 𝜃

Question 31

1 − 𝑠𝑖𝑛 2𝜃 =

Answer 31

(𝑠𝑖𝑛 𝜃 − 𝑐𝑜𝑠 𝜃)^2

Question 32

1 + 𝑠𝑖𝑛 2𝜃 =

Answer 32

(𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃)^2

Question 33

1 − 𝑐𝑜𝑠 2𝜃 / 1 + 𝑐𝑜𝑠 2𝜃 =

Answer 33

𝑡𝑎𝑛^2 𝜃

Question 34

2 𝑠𝑖𝑛^2 𝜃 =

Answer 34

1 − 𝑐𝑜𝑠 2𝜃

Question 35

2 𝑐𝑜𝑠^2 𝜃 =

Answer 35

1 + 𝑐𝑜𝑠 2𝜃

Question 36

𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 =

Answer 36

1/2 𝑠𝑖𝑛 2𝜃

Question 37

𝑠𝑖𝑛 2𝜃 =
(T)

Answer 37

2 𝑡𝑎𝑛 𝜃 / 1 + 𝑡𝑎𝑛^2 𝜃

Question 38

𝑐𝑜𝑠 2𝜃 =
(T)

Answer 38

1 − 𝑡𝑎𝑛^2 𝜃 / 1 + 𝑡𝑎𝑛^2 𝜃

Question 39

𝑡𝑎𝑛 2𝜃 =
(T)

Answer 39

2 𝑡𝑎𝑛 𝜃 / 1 + 𝑡𝑎𝑛^2 𝜃

Question 40

√1 + 𝑠𝑖𝑛 2𝜃 =

Answer 40

|𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃|

Question 41

√1 − 𝑠𝑖𝑛 2𝜃 =

Answer 41

|𝑠𝑖𝑛 𝜃 − 𝑐𝑜𝑠 𝜃|

Question 42

𝑠𝑖𝑛 3𝜃 =

Answer 42

3 𝑠𝑖𝑛 𝜃 − 4 𝑠𝑖𝑛^3 𝜃

Question 43

𝑐𝑜𝑠 3𝜃 =

Answer 43

4 𝑐𝑜𝑠^3 𝜃 − 3 𝑐𝑜𝑠 𝜃

Question 44

𝑡𝑎𝑛 3𝜃 =

Answer 44

3 𝑡𝑎𝑛 𝜃 − 𝑡𝑎𝑛^3 𝜃 / 1 − 3 𝑡𝑎𝑛^2 𝜃

Question 45

𝑠𝑖𝑛^3 𝜃 =

Answer 45

1/4 (3 𝑠𝑖𝑛 𝜃 − 𝑠𝑖𝑛 3𝜃)

Question 46

𝑐𝑜𝑠^3𝜃 =

Answer 46

1/4 (3 𝑐𝑜𝑠 𝐴 + 𝑐𝑜𝑠 3𝐴)

Question 47

𝑠𝑖𝑛(60 − 𝐴) 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛(60 + 𝐴) =

Answer 47

1/4 𝑠𝑖𝑛 3𝐴

Question 48

𝑐𝑜𝑠(600 − 𝐴) 𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠(600 + 𝐴)

Answer 48

1/4 𝑐𝑜𝑠 3𝐴

Question 49

𝑡𝑎𝑛(600 − 𝐴) 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛(600 + 𝐴) =

Answer 49

𝑡𝑎𝑛 3A