Trigonometric ratios of special angles worksheet with answers pdf

Problem 1 :

Evaluate sin 45° + cos 45°.

Problem 2 :

Evaluate sin 60° tan 30°.

Problem 3 :

Evaluate tan 45°/(tan 30° + tan 60°).

Problem 4 :

Evaluate tan2 60° - 2tan2 45° - cot2 30° + 2sin2 30.

Problem 5 :

Evaluate 4 (sin4 30° + cos460°) - 3 (cos245° - sin290°).

Problem 6 :

Evaluate 6 cos290° + 3 sin290° + 4 tan245°. 

Problem 7 :

Evaluate 4 cot245  - sec260 + sin260 + cos260.

Problem 8 :

Evaluate sin 30°cos 60° + cos 30° sin 60°.

Detailed Answer Key

Problem 1 :

Evaluate sin 45° + cos 45°.

Solution :

sin 45°  =  1/√2

cos 45°  =  1/√2

sin 45° + cos 45°  =  (1/√2)  +  (1/√2)

  =  (1 + 1)/√2

  =  2/√2

  =  (√2 ⋅ √2) / √2

  =  √2

Problem 2 :

Evaluate sin 60° tan 30°.

Solution :

sin 60°  =  √3/2

tan 30°  =  1/√3

sin 60° cos 30°  =  (√3/2) ⋅ (1/√3)

  =  1/2

Problem 3 :

Evaluate tan 45°/(tan 30° + tan 60°).

Solution :

tan 45°  =  1

tan 30°  =  1/2

tan 60°  =  √3

tan 45°/(tan 30° + tan 60°)  =  1/[(1/√3) + √3]

  =  1/[(1 + 3)/√3]

  =  √3/4

Problem 4 :

Evaluate tan2 60° - 2tan2 45° - cot2 30° + 2sin2 30.

Solution :

tan2 60°  =  (tan 60°)2   =  (√3)2  =  3

tan2 45°  =  (tan 45°)2   =  (1)2  =  1

cot2 30°  =  (cot 30°)2   =  (√3)2  =  3

sin2 30°  =  (sin 30°)2   =  (1/2)2  =  1/4

  =  3 - 2 (1) - 3 + 2(1/4)

  =  -2 + 1/2

  =  (-4 + 1)/2  =  -3/2

Problem 5 :

Evaluate 4 (sin4 30° + cos460°) - 3 (cos245° - sin290°).

Solution :

sin4 30°  =  (sin 30°)4   =  (1/2)4  =  1/16

cos4 60°  =  (cos 60°)4   =  (1/2)4  =  1/16

cos2 45°  =  (cos 45°)2   =  (1/√2)2  =  1/2

sin2 90°  =  (sin 90°)2   =  (1)2  =  1

  =  4 [(1/16) + (1/16)] - 3[(1/2) - 1]

  =  4(2/16)  -  3 (-1/2)

  =  (1/2) + (3/2)

  =  (1 + 3)/2  =  4/2  = 2

Problem 6 :

Evaluate 6 cos290° + 3 sin290° + 4 tan245°.

Solution :

cos290°  =  (cos 90°)2  =  (0)2  =  0

sin290°  =  (sin 90°)2  =  (1)2  =  1

tan245°  =  (tan 45°)2  =  (1)2  =  1

6 cos290° + 3 sin290° + 4 tan245°  =  6(0) + 3(1) + 4(1)

  =  0 + 3 + 4

  =   7

Problem 7 :

Evaluate 4 cot245  - sec260 + sin260 + cos260.

Solution :

cot245°  =  (cot 45°)2  =  (1)2  =  1

sec260  =  (sec 60°)2  =  (2)2  =  4

sin260  =  (sin 60°)2  =  (√3/2)2  =  3/4

cos260  =  (cos 60°)2  =  (1/2)2  =  1/4

  =  4(1) - 4 + (3/4) + (1/4)

  =  4 - 4 + (3+1)/4

  =  4/4  =  1

Problem 8 :

Evaluate sin 30°cos 60° + cos 30° sin 60°.

Solution :

sin 30°  =  1/2

cos 60°  =  1/2

cos 30°  =  √3/2

sin 60°  =  √3/2

  =  (1/2) (1/2) + (√3/2)(√3/2)

  =  (1/4) + (3/4)

  =  (1 + 3)/4

  =  4/4

  =  1

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