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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