determine an exact expression for the trigonometric function sin(13pi/12).
To determine an exact expression for the trigonometric function sin(13π/12), you need to know the exact values of sine for common angles such as 0°, 30°, 45°, 60°, and 90°.
Here's how you can find it:
1. Start with the given angle: 13π/12.
2. Convert the angle to degrees using the fact that π radians is equal to 180 degrees. So, divide 13π/12 by π and multiply by 180 to get the degrees:
(13π/12) * (180/π) = 195°
3. Now, you need to find an associated angle in the range of [0°, 360°]. Add or subtract multiples of 360° to get an angle within this range. In this case, you can subtract a multiple of 360° since the angle is already positive:
195° - 360° = -165°
4. Find the coterminal angle within the range of [0°, 360°] by adding 360°:
-165° + 360° = 195°
5. Determine which quadrant the angle lies in. In this case, 195° lies in the third quadrant.
6. Identify a reference angle, which is the acute angle formed between the terminal side of the angle and the x-axis. Find it by subtracting the angle from 180°:
180° - 195° = -15°
7. Determine the sign of the trigonometric function based on the quadrant that the angle lies in. In the third quadrant, sine is negative.
8. Use the reference angle (-15°) to determine the value of sin(-15°). Remember that sin is an odd function, meaning that sin(-x) = -sin(x).
sin(-15°) = -sin(15°)
9. Find the exact value of sin(15°).
You can use the special right triangle for the 30-60-90° triangle. In this triangle, the sine of 30° is equal to 1/2.
sin(15°) = sin(30°/2) = √((1 - cos(30°)) / 2)
= √((1 - √3/2) / 2)
= √((2 - √3) / 4)
= (√2 - √6) / 4
10. Substitute the value of sin(15°) back into the original expression for sin(13π/12):
sin(13π/12) = -sin(15°)
= -(√2 - √6) / 4
Therefore, the exact expression for sin(13π/12) is -(√2 - √6) / 4.
To determine the exact expression for the trigonometric function sin(13π/12), we can use the angle sum formula for sine:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
In this case, we can break down the angle 13π/12 as follows:
13π/12 = π/4 + 3π/6
Now, let's substitute this value into the angle sum formula:
sin(13π/12) = sin(π/4 + 3π/6)
Using the values of sin(π/4) and sin(3π/6) from the unit circle, we have:
sin(13π/12) = (sqrt(2)/2 * sqrt(3)/2) + (1/2 * 1/2)
simplifying, we get:
sin(13π/12) = sqrt(6)/4 + 1/4
Therefore, the exact expression for sin(13π/12) is (sqrt(6) + 1)/4.
note that 13pi/12 is also 195 degrees, which can be expressed as difference of two angles, 240 - 45:
sin(13pi/12) = sin(195) = sin(240-45)
*expanding this,
sin(240)cos(45)-cos(240)sin(45)
(-sqrt(3)/2)(sqrt(2)/2)-(-1/2)(sqrt(2)/2)
(-sqrt(6)/4)+(sqrt(2)/4)
therefore:
{[sqrt(2)-sqrt(6)]/4}
so there,, :)