Find exact value of the expression.
sin( 27pi/4 + 17pi/6)
sin( 27pi/4 + 17pi/6)
= sin(1215° + 510°) --- sometimes I think better with degrees
= sin(1725°)
= sin(285°) ---- 1725 = 4*360 + 285
= - sin 75°
=- sin(45°+30°)
= - [sin45cos30 + cos45sin30]
= - [ (1/√2)(√3/2) + (1/√2)(1/2)]
= - (√3 + 1)/(2√2)
or after rationalizing ...
= -(√6 + √2)/4
To find the exact value of the expression sin(27π/4 + 17π/6), you can use the trigonometric identity for the sum of angles:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
In this case, a = 27π/4 and b = 17π/6. Let's find the values for sin(27π/4) and sin(17π/6) separately.
First, let's convert 27π/4 to degrees. Since there are 2π radians in a complete revolution and 360 degrees in a complete revolution, we can find the equivalent value in degrees:
(27π/4) * (180°/π) = 27 * (180°/4) = 27 * 45° = 1215°
So, sin(27π/4) is the same as sin(1215°).
Next, let's convert 17π/6 to degrees:
(17π/6) * (180°/π) = 17 * (180°/6) = 17 * 30° = 510°
So, sin(17π/6) is the same as sin(510°).
Now, using the sum of angles formula, we can calculate sin(27π/4 + 17π/6):
sin(27π/4 + 17π/6) = sin(1215° + 510°)
Since the sine function has a periodicity of 360°, we can subtract 360° from the angle until it falls within the range of 0° to 360°.
(1215° + 510°) - 360° = 1365° - 360° = 1005°
So, sin(27π/4 + 17π/6) is the same as sin(1005°).
Now, we need to find the reference angle, which is the smallest angle between the terminal side and the x-axis. In this case, it is 1005° - 360° = 645°.
The sine function is positive in the first and second quadrants, so we take the sine of the reference angle:
sin(645°) = sin(645° - 360°) = sin(285°)
Finally, we need to find the exact value of sin(285°). You can use the unit circle or a calculator to find that:
sin(285°) ≈ -0.266
Therefore, the exact value of the expression sin(27π/4 + 17π/6) is approximately -0.266.