sin 19pi/12
Perhaps you can visualize degrees better than radians?
19π/12 radians = 285°
285° is in quadrant IV, so
sin 19π/12 = sin 285°
= - sin(75°) , due to the CAST rule
= - sin(45 + 30)
= -(sin45cos30 + cos45sin30)
= - (√2/2(√3/2) + √2/2(1/2)
= -√6/4 - √2/4
or -(√6 + √2)/4
To find the value of sin(19π/12), we can use the unit circle and the properties of trigonometric functions.
Step 1: Understanding the angle
The given angle is 19π/12. To understand this angle, let's convert it into degrees first. We know that π radians is equal to 180 degrees. Thus, we can convert the given angle as follows:
19π/12 * (180 degrees/π) = 285 degrees
So, sin(19π/12) is equivalent to sin(285 degrees).
Step 2: Using the unit circle
We can find the value of sin(285 degrees) using the unit circle.
Start by drawing a unit circle, where the radius is 1 unit long. The center of the unit circle is the origin (0,0).
Next, find the angle of 285 degrees. Since it is more than 180 degrees, we know that it is in the third quadrant.
In the third quadrant, the value of sin is always negative. So, we need to find the negative value of sin(285 degrees).
Now, locate the point on the unit circle that intersects with the angle of 285 degrees. Draw a line from that point to the x-axis, creating a right triangle.
Step 3: Applying trigonometric properties
The x-coordinate of the point where the line intersects the x-axis is the cosine of the angle, while the y-coordinate is the sine of the angle.
Since we are looking for the sine, we focus on the y-coordinate of the point.
In the right triangle, the hypotenuse is the radius of the unit circle (1 unit), and the y-coordinate is the opposite side, while the x-coordinate is the adjacent side.
Using the Pythagorean theorem, we can find the length of the opposite side:
opposite^2 + adjacent^2 = hypotenuse^2
opposite^2 + 1^2 = 1^2
opposite^2 = 1 - 1 = 0
opposite = 0
Therefore, the value of sin(285 degrees) is 0.
So, sin(19π/12) = sin(285 degrees) = 0.