In Trigonometry, you usually memorize the standard angles (0°, 30°, 45°, 60°, 90°). But what happens if an exam asks for the exact value of cos(15°)? You cannot find it directly on the standard table, but you can easily calculate it using a compound angle formula.
Because of the complementary angle rule in trigonometry, cos(15°) is exactly equal to sin(75°)!
The exact value of cos(15°) in fraction/surd form is: cos(15°) = (√6 + √2) / 4
(In decimal form, it is approximately 0.9659).
To find this value, we express 15° as the difference between two standard angles that we already know: 45° and 30°. (Since 45 - 30 = 15).
Step 1: The Formula Use the cosine subtraction formula: cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Step 2: Substitute the angles Let A = 45° and B = 30°. cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°)
Step 3: Insert standard table values We know that:
Step 4: Calculate cos(15°) = (1/√2) × (√3/2) + (1/√2) × (1/2) cos(15°) = (√3 / 2√2) + (1 / 2√2)
Since the denominators are the same, combine the numerators: cos(15°) = (√3 + 1) / 2√2
While the answer above is correct, mathematicians prefer not to leave square roots in the denominator. We multiply the top and bottom by √2:
= [ (√3 + 1) × √2 ] / [ 2√2 × √2 ] = (√6 + √2) / 4
You use the exact same logic, but with the sine subtraction formula: sin(A-B) = sin(A)cos(B) - cos(A)sin(B). The answer will be **(√6 - √2) / 4**.
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