Worksheet: Evaluating Trigonometric Functions Using Pythagorean Identities

In this worksheet, we will practice using the Pythagorean identities to find the values of trigonometric functions.

Q1:

Find c o s 𝜃 given s i n 𝜃 = 3 5 where 2 7 0 𝜃 < 3 6 0 .

  • A 3 4
  • B 4 5
  • C 3 5
  • D 4 5

Q2:

Find c o t 𝜃 given s i n 𝜃 = 3 5 where 9 0 < 𝜃 < 1 8 0 .

  • A 4 3
  • B 3 4
  • C 4 3
  • D 3 4

Q3:

Find the value of c o t 𝜃 given c s c 𝜃 = 2 5 9 .

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

Q4:

Find the value of s i n 𝜃 given c o s 𝜃 = 2 1 2 9 where 9 0 < 𝜃 < 1 8 0 .

  • A 2 0 2 9
  • B 2 0 2 9
  • C 2 0 2 1
  • D 2 1 2 9
  • E 2 0 2 1

Q5:

Find the value of s i n c o s 𝜃 𝜃 given s i n c o s 𝜃 + 𝜃 = 5 4 .

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

Q6:

Find the value of 2 𝜃 𝜃 s i n c o s given 1 2 𝜃 + 5 = 0 t a n where 1 8 0 < 𝜃 < 3 6 0 .

  • A 1 2 0 1 6 9
  • B 1 2 0 1 6 9
  • C 5 2 4
  • D 5 2 4

Q7:

Find s e c t a n 𝜃 𝜃 given s e c t a n 𝜃 + 𝜃 = 1 4 2 7 .

  • A 2 7 1 4
  • B 2 7 1 4
  • C 4 1 1 4
  • D 4 1 1 4

Q8:

Find the value of s e c 𝜃 given s e c t a n 𝜃 𝜃 = 1 6 where 0 < 𝜃 < 𝜋 2 .

  • A 2 0 9
  • B 3 7 1 8
  • C 3 5 1 2
  • D 3 7 1 2

Q9:

Knowing that s i n 𝑥 = 1 3 7 and 𝜋 2 𝑥 𝜋 , find t a n 𝑥 .

  • A 1 3 6
  • B 6 1 3
  • C 1 3 6
  • D 1 3 3 6
  • E 6 1 3

Q10:

Find the value of t a n ( 3 6 0 𝜃 ) given c o t 𝜃 = 4 3 where 0 < 𝜃 < 9 0 .

  • A 4 3
  • B 3 4
  • C 3 4
  • D 4 3

Q11:

Find the value of t a n ( 1 8 0 + 𝜃 ) given s i n 𝜃 = 3 5 where 0 < 𝜃 < 9 0 .

  • A 4 5
  • B 4 5
  • C 3 4
  • D 3 4

Q12:

Find c s c s e c 𝑎 𝑎 given c o s s i n 𝑎 𝑎 = 2 7 .

  • A 1 4 4 5
  • B 2 8 4 5
  • C 4 5 1 4
  • D 2 8 4 5

Q13:

Simplify s i n s i n ( 𝜋 𝜃 ) + ( 2 7 0 𝜃 ) .

Q14:

Simplify s i n s i n 𝜃 + ( 9 0 𝜃 ) .

Q15:

Find s i n 𝐴 , given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where c o s 𝐴 = 0 . 8 .

  • A 4 5
  • B 5 3
  • C 4 3
  • D 3 5
  • E 5 4

Q16:

Find 1 + 𝐴 t a n , given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐶 where 𝐴 𝐵 = 1 0 c m and 𝐵 𝐶 = 6 c m .

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

Q17:

Find the value of c s c s i n t a n c s c 𝜃 𝜃 𝜃 𝜃 given 𝜃 𝜋 2 , 𝜋 and c o s 𝜃 = 4 5 .

  • A 1 4
  • B 1 4
  • C 9 4
  • D 9 4

Q18:

Find the value of c s c s i n t a n c o t c o s 𝜃 𝜃 𝜃 𝜃 + 𝜃 given 𝜃 𝜋 , 3 𝜋 2 and s i n 𝜃 = 4 5 .

  • A 5 9 2 5
  • B 5 9 2 5
  • C 9 2 5
  • D 9 2 5

Q19:

Find the value of s e c ( 𝜃 ) given c s c 𝜃 = 1 3 5 where 0 < 𝜃 < 9 0 .

  • A 1 2 1 3
  • B 1 3 1 2
  • C 1 2 1 3
  • D 1 3 1 2

Q20:

Find the value of 1 7 𝜃 + 9 𝜃 + 8 𝜃 s i n c o s s e c .

Q21:

Find the value of s i n c o s c o s s i n 𝛼 𝛽 𝛼 𝛽 , given t a n 𝛼 = 3 4 where 𝛼 is the smallest positive angle and t a n 𝛽 = 1 5 8 where 1 8 0 < 𝛽 < 2 7 0 .

  • A 3 6 8 5
  • B 1 3 8 5
  • C 1 3 8 5
  • D 3 6 8 5

Q22:

Find t a n 𝜃 given s i n 𝜃 = 3 5 where 2 7 0 𝜃 < 3 6 0 .

  • A 4 3
  • B 3 4
  • C 3 4
  • D 4 3

Q23:

Find the value of s e c ( 1 , 0 8 0 + 𝜃 ) given 4 𝜃 3 = 0 t a n where 0 < 𝜃 < 1 8 0 .

  • A 5 4
  • B 5 4
  • C 5 3
  • D 5 3

Q24:

Find the value of 3 5 𝛼 1 6 𝛼 s i n c o t given c o s 𝛼 = 9 2 5 where 1 8 0 < 𝛼 < 2 7 0 .

Q25:

Find the value of t a n 𝜃 given c s c 𝜃 = 5 3 where 1 8 0 < 𝜃 < 2 7 0 .

  • A 4 5
  • B 3 4
  • C 4 5
  • D 3 4
  • E 3 5

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