I need help on these problems: Find the exact value of tan(23pi/6)
also, Find the reference angle in radians and degrees for (34pi/18)
please help walk me through these.
tangent has a period of π
so
tan(x) = tan(x+π) = tan(x + 2π) ...
Also, tan(x)=-tan(π - x)
so
tan(23π/6)
=tan(5π/6 + 3π)
=tan(5π/6)
=-tan(π/6)
=-√3/3
A reference angle is the acute angle between the terminal side of the angle and the x-axis.
So the reference angle of 34π/18 is 2π/18 = π/9 radians = &pi/9 *(180/&pi)° = 20°
Certainly! Let's go step by step through each problem:
1. To find the exact value of tan(23π/6), we need to determine the angle associated with this trigonometric function.
The angle in question, 23π/6, is more than a complete revolution (2π), which means we need to bring it back to a standard position.
To do that, we can subtract 2π from the original angle until we get within one revolution. In this case, let's subtract 2π:
23π/6 - 2π = 23π/6 - 12π/6 = 11π/6
Now, we have 11π/6, which is within one revolution.
Next, we need to determine the reference angle. The reference angle is the positive acute angle formed between the x-axis and the terminal side of the angle.
To find the reference angle, subtract the angle from a full revolution (2π):
2π - 11π/6 = 12π/6 - 11π/6 = π/6
So, the reference angle in radians is π/6.
Finally, we can find the exact value of tangent using the reference angle. For tangent, we use the following trigonometric identity:
tan(π/6) = sin(π/6) / cos(π/6)
The values for sine and cosine of π/6 are √3/2 and 1/2, respectively, based on the unit circle. By substituting these values in, we get:
tan(π/6) = (√3/2) / (1/2) = (√3/2) * (2/1) = √3
Therefore, the exact value of tan(23π/6) is √3.
2. For the second problem, we are asked to find the reference angle in radians and degrees for (34π/18).
Similar to the previous problem, we need to bring the angle back to a standard position within one revolution.
34π/18 - 2π = 34π/18 - 36π/18 = -2π/18
Now, we have -2π/18, which is within one revolution.
To find the reference angle, we need to take the absolute value:
|-2π/18| = 2π/18
The reference angle in radians is 2π/18.
To convert the reference angle to degrees, we can use the fact that π radians is equal to 180 degrees:
2π/18 * (180/π) = 20 degrees
So, the reference angle in degrees is 20.
I hope this helps! Let me know if you have any further questions.