Find an exact value for csc 11pi/12
How do I do this?
I change it to 1/(sin 11pi/12)
but then I'm stuck what to do..
Thanks
11π / 12 corresponds to a 15º reference angle in Q II
15º is half of 30º
use a half-angle identity to find the exact value
photomath
To find the exact value of csc (11π/12), you can follow these steps:
1. Start with the equation csc θ = 1/sin θ.
2. In this case, θ is 11π/12.
3. First, you need to find the value of sin (11π/12).
4. To do this, convert 11π/12 to degrees: (11π/12) * (180/π) = 165°.
5. Next, we need to determine the quadrant in which 165° lies. Since 0 < 165° < 180°, it is in the second quadrant.
6. In the second quadrant, sin θ is positive. Therefore, sin 165° > 0.
7. Now, we can find the value of sin 165°.
8. One way to determine the exact value of sin 165° is by using the sine addition formula: sin (A + B) = sin A * cos B + cos A * sin B.
9. In this case, we can use the angle sum identity sin (π/3 + π/4) = sin (7π/12) = sin (π/3) * cos (π/4) + cos (π/3) * sin (π/4).
10. From this identity, we know that sin (π/3) = √3/2 and cos (π/4) = √2/2.
11. Plugging in these values, we have sin (7π/12) = (√3/2) * (√2/2) + (1/2) * (√2/2) = (√6 + √2) / 4.
12. Therefore, sin (11π/12) = sin 165° = (√6 + √2) / 4.
13. Finally, substitute this value back into the original equation: csc (11π/12) = 1/sin (11π/12) = 1 / [ (√6 + √2) / 4 ].
14. To simplify further, multiply the numerator and denominator by 4: csc (11π/12) = 4 / (√6 + √2).
15. To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator: csc (11π/12) = (4 / (√6 + √2) ) * (√6 - √2) / (√6 - √2).
16. Expanding the denominator using the difference of squares, we get: csc (11π/12) = (4(√6 - √2)) / (6 - 2) = 2(√6 - √2).
Therefore, the exact value of csc (11π/12) is 2(√6 - √2).