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Worksheet: Trigonometric Identities

Q1:

For any π‘₯ β‰  π‘˜ πœ‹ ( π‘˜ ∈ β„€ ) , what is 1 + π‘₯ c o t 2 ?

  • A s i n 2 π‘₯
  • B 1 π‘₯ c o s 2
  • C c o s 2 π‘₯
  • D 1 π‘₯ s i n 2
  • E 1 π‘₯ s i n

Q2:

Knowing that c o s π‘₯ = √ 1 7 5 and that 3 πœ‹ 2 ≀ π‘₯ ≀ 2 πœ‹ , find s i n π‘₯ .

  • A 2 √ 2 5
  • B βˆ’ 8 2 5
  • C 8 2 5
  • D βˆ’ 2 √ 2 5
  • E 5 βˆ’ √ 1 7 5

Q3:

For any π‘₯ β‰  πœ‹ 2 + π‘˜ πœ‹ ( π‘˜ ∈ β„€ ) , what is c o s 2 π‘₯ ?

  • A 1 + π‘₯ t a n 2
  • B 1 1 + π‘₯ c o t 2
  • C 1 + π‘₯ c o t 2
  • D 1 1 + π‘₯ t a n 2
  • E 1 π‘₯ t a n 2

Q4:

Find the value of t a n πœƒ given s e c t a n πœƒ βˆ’ πœƒ = 2 1 3 where 0 < πœƒ < πœ‹ 2 .

  • A 3 7 2 1 6 9
  • B 1 7 3 5 2
  • C 3 4 6 1 6 9
  • D 1 6 5 5 2

Q5:

Find the value of 7 ο€Ή πœƒ βˆ’ πœƒ  t a n s e c 2 2 2 .

Q6:

Consider point 𝑀 ( π‘₯ , 𝑦 ) on the unit circle that represents angle πœƒ , where 0 ≀ πœƒ ≀ πœ‹ 2 . Additional points on the unit circle are 𝑁 ( βˆ’ π‘₯ , 𝑦 ) , 𝑃 ( βˆ’ π‘₯ , βˆ’ 𝑦 ) , and 𝑄 ( π‘₯ , βˆ’ 𝑦 ) . 𝑋 2 is a point with coordinates ( π‘₯ , 0 ) and π‘Œ 1 a point with coordinates ( 0 , 𝑦 ) .

Which of the following is NOT true?

  • A 𝑋 𝑀 = 𝑂 π‘Œ 2 1
  • B 𝑂 𝑋 + 𝑋 𝑀 = 1 2 2 2 2
  • C c o s πœƒ = π‘₯ , s i n πœƒ = 𝑦
  • D s i n c o s πœƒ + πœƒ = 1
  • E c o s s i n 2 2 πœƒ + πœƒ = 1

Which of the following is NOT true?

  • A π‘Œ π‘Œ = π‘Œ 𝑀 1 2 1
  • BThere are 4 triangles in the figure that are congruent to β–³ 𝑂 𝑀 𝑋 2 .
  • C | πœƒ | = | ( πœ‹ βˆ’ πœƒ ) | = | ( πœ‹ + πœƒ ) | = | ( 2 πœ‹ βˆ’ πœƒ ) | c o s c o s c o s c o s
  • D s i n s i n s i n s i n 2 2 2 2 πœƒ = ( πœ‹ βˆ’ πœƒ ) = ( πœ‹ + πœƒ ) = ( 2 πœ‹ βˆ’ πœƒ )
  • EThere are 4 triangles in the figure that are congruent to β–³ 𝑂 𝑀 π‘Œ 1 .

Over which interval is c o s s i n 2 2 πœƒ + πœƒ = 1 true?

  • A ο€» 0 , πœ‹ 2 
  • B  0 , πœ‹ 2 
  • CIt is always true.
  • D  0 , πœ‹ 2 
  • E ο€» 0 , πœ‹ 2 

Q7:

Knowing that c o s π‘₯ = βˆ’ 3 5 and that πœ‹ ≀ π‘₯ ≀ 3 πœ‹ 2 , find t a n π‘₯ .

  • A 5 3
  • B 1 6 9
  • C 2 5 9
  • D 4 3
  • E 3 4

Q8:

Knowing that t a n π‘₯ = 2 √ 1 9 and 0 ≀ π‘₯ ≀ πœ‹ 2 , find s i n π‘₯ .

  • A 7 6 7 7
  • B 1 7 7
  • C 1 √ 7 7
  • D 2 √ 1 9 √ 7 7
  • E 1 √ 7 6

Q9:

Find the value of c s c 2 πœƒ given c o t πœƒ = 7 9 .

  • A 2 5 6 8 1
  • B 1 7 9
  • C 1 6 9
  • D 1 4 9 8 1

Q10:

Knowing that s i n π‘₯ = βˆ’ √ 7 4 and 3 πœ‹ 2 ≀ π‘₯ ≀ 2 πœ‹ , find c o s π‘₯ .

  • A βˆ’ 3 4
  • B βˆ’ 9 1 6
  • C 9 1 6
  • D 3 4
  • E 4 + √ 7 4

Q11:

Find the value of , given where and where .

  • A 1
  • B
  • C
  • D

Q12:

Find the value of s i n c o s πœƒ πœƒ given s i n c o s s i n c o s 2 2 πœƒ βˆ’ πœƒ πœƒ βˆ’ πœƒ = 5 3 .

  • A 1 7 9
  • B 1 6 9
  • C 3 2 9
  • D 8 9

Q13:

Find the value of given and .

  • A
  • B
  • C
  • D

Q14:

Find given where .

Q15:

Find the value of t a n c o t 2 2 πœƒ + πœƒ given t a n c o t πœƒ + πœƒ = 1 7 .

Q16:

Find the value of s e c c s c 𝑋 + 𝑋 given s i n c o s 𝑋 + 𝑋 = βˆ’ 6 7 where πœ‹ 2 < 𝑋 < πœ‹ .

  • A βˆ’ 7 6
  • B 1 3 9 8
  • C 4 2 1 3
  • D 8 4 1 3

Q17:

Find, without using a calculator, the value of s i n c o s 2 𝐡 2 2 𝐡 given c o s 𝐡 = 4 5 where 3 πœ‹ 2 < 𝐡 < 2 πœ‹ .

  • A βˆ’ 1 2 2 5
  • B βˆ’ 6 7
  • C βˆ’ 6 2 5
  • D βˆ’ 1 2 7

Q18:

Simplify ( π‘Ž βˆ’ 1 ) 1 βˆ’ π‘Ž t a n t a n   .

  • A t a n 2 π‘Ž
  • B s e c 2 π‘Ž
  • C 1 βˆ’ 2 π‘Ž t a n
  • D s e c t a n 2 π‘Ž βˆ’ 2 π‘Ž

Q19:

Find the value of t a n 2 𝑋 given t a n t a n 𝑋 1 βˆ’ 𝑋 = 2 2 .

  • A 1 2
  • B 2
  • C 1
  • D 4

Q20:

Find c s c c o t πœƒ + πœƒ given c s c c o t πœƒ βˆ’ πœƒ = 4 5 .

  • A 1 4
  • B 5 2
  • C 1 2
  • D 5 4