Question

Trigonometric Ratios: "First steps in trigonometry"

Answer 1

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Answer:

  sin: 0, (√(2-√3))/2, 1/2, (√2)/2, (√3)/2, (√(2+√3))/2, 1

  cos are the same values in reverse order

Step-by-step explanation:

Given the side ratios of the "special" right triangles, we know that ...

  sin(30°) = cos(60°) = 1/2

  sin(45°) = cos(45°) = 1/√2 = (√2)/2

  sin(60°) = cos(30°) = (√3)/2

Using the half-angle formulas in the second attachment, we can find exact values for the 15° and 75° angles.

  sin(15°) = sin(30°/2) = √((1 -cos(30°)/2) = √((1 -(√3)/2)/2) = (√(2 -√3))/2

  cos(15°) = √((1 +cos(30°)/2) = √((1 +(√3)/2)/2) = (√(2 +√3))/2

Then the exact table values are ...

• sin(0°) = cos(90°) = 0
• sin(15°) = cos(75°) = (√(2-√3))/2
• sin(30°) = cos(60°) = 1/2
• sin(45°) = cos(45°) = (√2)/2
• sin(60°) = cos(30°) = (√3)/2
• sin(75°) = cos(15°) = (√(2+√3))/2
• sin(90°) = cos(0°) = 1

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Additional comments

The numerical sine and cosine values in the attachment are computed by the spreadsheet functions SIN( ) and COS( ). Those functions require an argument in radians. The RADIANS( ) function converts from degrees to radians.

The "special" right triangles are the isosceles right triangle with side ratios 1:1:√2, and the 30°-60°-90° triangle with side ratios 1:√3:2.