3. Find the exact value of the following, show all work.
a. sin^2 (75^0)/sin45, cos^2, 30^0
b. tan(135) + sin(30)cot(120)
a. To find the exact values of trigonometric functions, we can use the values of common reference angles (0°, 30°, 45°, 60°, 90°) and trigonometric identities.
a.
1. Start by finding the values of the angles involved:
- 75° is not a common reference angle, but we can use the difference identity for sine to express it in terms of a common reference angle:
sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°)
- We know sin(45°) = cos(45°) = √2/2 and sin(30°) = 1/2, cos(30°) = √3/2.
2. Plug in the values into the expression sin^2(75°)/sin(45°):
sin^2(75°)/sin(45°) = [sin(45°)cos(30°) + cos(45°)sin(30°)]^2/ sin(45°)
3. Simplify the expression:
sin^2(75°)/sin(45°) = [(√2/2)(√3/2) + (√2/2)(1/2)]^2/ [√2/2]
= [(√6/4) + (√2/4)]^2 / √2
= [ (√6 + √2)/4 ]^2 / [ (√2)/2 ]
= [ (√6 + √2)^2 / 16 ] * [ 2 / √2 ]
= [ 6 + 2√12 + 2 + 2√2] / 16 * √2 / 2
= [8 + 2√2 + 2√6] / [16 * √2 / 2]
= [8 + 2√2 + 2√6] / [ 8√2 ]
= (8 + 2√2 + 2√6) / (8√2)
= (8/8√2) + (2√2/8√2) + (2√6 / 8√2)
= 1/√2 + 1/√2 + ( 2√6 / 8√2)
= 2/√2 + 2√6 / 8√2
= √2 + √6 / 4√2
= (√2 + √6) / ( 4 * √2 )
= (√2 + √6) / ( 4 * √2 ) * √2/√2
= (√2√2 + √6 * √2 ) / ( 4 * 2 )
= (2 + √12) / 8
= (2 + 2√3) / 8
= (1 + √3) / 4
Hence, the exact value of sin^2 (75°)/sin45° is (1 + √3) / 4.
b.
To find the exact value of the expression tan(135°) + sin(30°)cot(120°), we can use the values of common reference angles and trigonometric identities:
1. Start by finding the values of the angles involved:
- tan(135°) can be expressed in terms of a common reference angle using the tangent's periodicity and symmetry properties:
tan(135°) = tan(180° - 45°) = -tan(45°) = -1
- sin(30°) = 1/2
- cot(120°) can be expressed as the reciprocal of the tangent: cot(120°) = 1/tan(120°)
2. Plug in the values into the expression: tan(135°) + sin(30°)cot(120°)
tan(135°) + sin(30°)cot(120°) = -1 + (1/2) * (1/tan(120°))
3. Simplify the expression:
To find the exact value of tan(120°), we need to consider its reference angle, 60°.
- tan(60°) = √3
- cot(120°) = 1/tan(120°) = 1/(−√3) = −√3/3
Substituting these values, we have:
-1 + (1/2) * (−√3/3) = -1 - (√3/6)
Therefore, the exact value of tan(135°) + sin(30°)cot(120°) is -1 - (√3/6).