EVALUATING TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES

About "Evaluating trigonometric functions of special angles"

Evaluating trigonometric functions of special angles :

Here we are going to see how to evaluate trigonometric functions of special angles.

To evaluate the given trigonometric expressions, we use the table given below.

θ

0°

30°

45°

60°

90°

sin θ

0

1/2

1/√2

√3/2

1

cos θ

1

√3/2

1/√2

1/2

0

tan θ

0

1/√3

1

√3

Evaluating trigonometric functions of special angles - Examples

Let us look into some examples given below.

Example 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

Example 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

Example 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

Example 4 :

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

Solution :

tan60°  =  (tan 60°)2   =  (√3) =  3

tan45°  =  (tan 45°)2   =  (1) =  1

cot30°  =  (cot 30°)2   =  (√3) =  3

sin30°  =  (sin 30°)2   =  (1/2) =  1/4

  =  3 - 2 (1) - 3 + 2(1/4)

  =  -2 + 1/2

  =  (-4 + 1)/2  =  -3/2

Example 5 :

Evaluate 4 (sin30° + cos460°) - 3 (cos245° - sin290°)

Solution :

sin30°  =  (sin 30°)4   =  (1/2) =  1/16

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

cos45°  =  (cos 45°)2   =  (1/2) =  1/2

sin90°  =  (sin 90°)2   =  (1) =  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

Example 6 :

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

Solution :

cos290°  =  (cos 90°)=  (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

Example 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

Example 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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