What is cos(11pi/3)? Explain your thinking.
What is sin(13pi/4) ? Explain your thinking.
3 2/3 pi
3 pi ends up at 180 degrees (2 pi + pi)
2/3 pi = 120 degrees
180 + 120 = 300 = 360 - 60
so 60 degrees below x axis in quadrant 4
cos -60 = 1/2
so
cos (11 pi/3) = 1/2
Now you do any more like that.
To find the value of cos(11π/3), we can use the unit circle to determine the cosine value at the specific angle.
1. Start with the angle 11π/3.
2. Recall that 2π is one full revolution around the unit circle. Therefore, we can subtract or add 2π to the angle until it falls within a range of 0 to 2π.
- 11π/3 is equivalent to (9π + 2π)/3, which simplifies to 3π + 2π/3.
- Since 3π is one full revolution, we can focus on finding the cosine value at 2π/3.
3. Locate the point on the unit circle that corresponds to the angle 2π/3.
- The angle 2π/3 falls in the second quadrant of the unit circle.
- In the second quadrant, the x-coordinate (cosine) is negative.
4. Therefore, cos(11π/3) is equal to cos(2π/3), which is the same as the x-coordinate of the point on the unit circle corresponding to the angle 2π/3.
- The x-coordinate (cosine value) at 2π/3 is -1/2.
5. Thus, cos(11π/3) = -1/2.
Similarly, to find the value of sin(13π/4), we can use the unit circle to determine the sine value at the specific angle.
1. Start with the angle 13π/4.
2. Recall that 2π is one full revolution around the unit circle. Therefore, we can subtract or add 2π to the angle until it falls within a range of 0 to 2π.
- 13π/4 is equivalent to (8π + 5π)/4, which simplifies to 2π + 5π/4.
- Since 2π is one full revolution, we can focus on finding the sine value at 5π/4.
3. Locate the point on the unit circle that corresponds to the angle 5π/4.
- The angle 5π/4 falls in the third quadrant of the unit circle.
- In the third quadrant, both the x-coordinate and the y-coordinate (sine) are negative.
4. Therefore, sin(13π/4) is equal to sin(5π/4), which is the same as the y-coordinate of the point on the unit circle corresponding to the angle 5π/4.
- The y-coordinate (sine value) at 5π/4 is -√2/2.
5. Thus, sin(13π/4) = -√2/2.
To find the value of cos(11π/3), we can use the unit circle and the properties of cosine function. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.
Step 1: Determine the reference angle
The reference angle is the acute angle formed between the positive x-axis and the terminal side of the angle in standard position. In this case, the angle is 11π/3. To find the reference angle, subtract multiples of 2π until the angle is in the range between 0 and 2π.
11π/3 - 6π = π/3
So the reference angle is π/3.
Step 2: Determine the quadrant
Since the reference angle π/3 falls within the first quadrant (0 to π/2), the cosine function will be positive.
Step 3: Evaluate cos(π/3)
In the first quadrant, the cosine value is positive. So cos(π/3) = cos(60°) = 1/2.
To find the value of sin(13π/4), we can follow a similar process using the unit circle and the properties of the sine function.
Step 1: Determine the reference angle
The reference angle is the acute angle formed between the positive x-axis and the terminal side of the angle in standard position. In this case, the angle is 13π/4. To find the reference angle, subtract multiples of 2π until the angle is in the range between 0 and 2π.
13π/4 - 8π/4 = 5π/4
So the reference angle is 5π/4.
Step 2: Determine the quadrant
Since the reference angle 5π/4 falls within the third quadrant (π to 3π/2), the sine function will be negative.
Step 3: Evaluate sin(5π/4)
In the third quadrant, the sine value is negative. So sin(5π/4) = sin(225°) = -√2/2.
Therefore, the value of sin(13π/4) is -√2/2.