EVALUATING TRIGONOMETRIC RATIOS FOR SPECIAL ANGLES

Example 1 :

Verify the following equality :

sin²60°+ cos²60°  =  1

Solution :

sin²60° + cos²60°  =  (√3/2)² + (1/2)²

sin²60° + cos²60°  =  (3/4) + (1/4)

  sin²60° + cos²60°  =  (3 + 1)/4 

 sin²60° + cos²60°  =  4/4

 sin²60° + cos²60°  =  1 

Example 2 :

Verify the following equality :

 1 + tan²30°  =  sec²30°

Solution :

1 + tan²30°  =  1 + (1/√3)²

1 + tan²30°  =  1 + (1/3)

  1 + tan²30°  =  4/3 -----(1)

sec²30°  =  (2/√3)²

sec²30°  =  4/3 -----(2)

From (1) and (2), we get

1 + tan²30°  =  sec²30°

Example 3 :

Verify the following equality :

cos90°  =  1 - 2 sin²45°  =  2 cos²45° - 1

Solution :

cos 90°  =  0 -----(1)

1 - 2 sin²45°  =  1 - 2(1/√2)² 

 1 - 2 sin²45°  =  1 - 2(1/2) 

1 - 2 sin²45°  =  0 -----(2)

2 cos² 45° - 1  =  2(1/√2)² - 1

2 cos² 45° - 1  =  2(1/2) - 1

2 cos² 45° - 1  =  0 -------(3)

From (1), (2) and (3), we get

cos90°  =  1 - 2 sin²45°  =  2 cos²45° - 1

Example 4 :

Verify the following equality :

sin 30° cos 60° + cos 30° sin 60°  =  sin 90°

Solution :

sin 30° cos 60° + cos 30° sin 60° : 

  =  (1/2) (1/2) + (√3/2) (√3/2) 

  =  (1/4) + (3/4)

  =  (1 + 3)/4

  =  4/4 

  =  1 -----(1)

sin 90°  =  1 -----(2)

From (1) and (2), we get

sin 30° cos 60° + cos 30° sin 60°  =  sin 90°

Example 5 :

Find the value of the following :

(tan45°/cosec30°) + (sec60°/cot45°) - (5sin90°/2cos0°)

Solution :

(tan45°/cosec30°) + (sec60°/cot45°) - (5sin90°/2cos0°) :

=  (1/2) + (2/1) - (5(1)/2(1))

  =  (1/2) + (4/2) - (5/2)

  =  (1 + 4 - 5)/2

  =  0

Example 6 :

Find the value of the following :

(sin 90°+cos 60°+cos 45°) × (sin 30°+cos 0°-cos 45°)

Solution :

(sin 90°+cos 60°+cos 45°) × (sin 30°+cos 0°-cos 45°) :

  =  [1 + (1/2) + (1/√2)] × [(1/2) + 1 - (1/√2)]

=  [(3/2) + (1/√2)] × [(3/2) - (1/√2)]

=  [(3/2)² - (1/√2)²]

=  [(9/4) - (1/2)]

=  (9 - 2)/4

=  7/4

Example 7 :

Find the value of the following :

sin²30° - 2 cos³60° + 3 tan⁴45°

Solution :

sin²30° - 2 cos³60° + 3 tan⁴45° :

=  (1/2)² - 2(1/2)³ + 3(1)⁴

=  (1/4) - (1/4) + 3

=  3

Example 8 :

Verify : 

cos 3A = 4 cos³A - 3 cos A , when A  = 30°

Solution :

cos 3A  =  cos 3(30°)

cos 3A  =  cos 90°

  cos 3A  =  0  ---(1)

4 cos³A - 3 cosA  =  4 cos³30° - 3 cos 30°

4 cos³A - 3 cosA  =  4 (√3/2)³ - 3 (√3/2)

4 cos³A - 3 cosA  =  (12√3/8) - (3√3/2)

4 cos³A - 3 cosA  =  (3√3/2) - (3√3/2)

  4 cos³A - 3 cosA  =  0 -----(2)

From (1) and (2), we get

cos 3A = 4 cos³A - 3 cos A

when A  = 30°.

Example 9 :

Find the value of 8sin 2x cos 4x sin6x, when x = 15°.

Solution :

8 sin 2x cos 4x sin6x :

=  8 sin 2(15°) cos 4(15°) sin 6(15°)

=  8 sin 30° cos 60° sin 90°

=  8(1/2) (1/2) (1)

=  2

Example 10 :

Find the value of the following :

tan²60° - 2 tan²45° - cot²30° + 2 sin²30° + (3/4)csc²45°

Solution :

tan²60° - 2 tan²45° - cot²30° + 2 sin²30° + (3/4)csc²45°:

=  (√3)² - 2(1)² - (√3)² + 2(1/2)² + (3/4)(√2)²

=  3 - 2(1) - 3 + 2(1/4) + (3/4)(2)

=  3 - 2 - 3 + 1/2 + 3/2

=  6/2 - 4/2 - 6/2 + 1/2 + 3/2

=  (6 - 4 - 6 + 1 + 3) / 2

=  0

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