determine the sign of tan(7π/6) and the quadrant in which it lies.
To determine the sign of tan(7π/6), we need to find the value of tan(7π/6).
First, let's find the reference angle for 7π/6 by subtracting it from π, which is the angle on the unit circle that lies on the x-axis in the positive direction:
Reference angle = π - 7π/6 = (6π/6) - (7π/6) = -π/6
Since the reference angle is -π/6, tan(-π/6) is negative.
Now, let's determine the quadrant in which 7π/6 lies. The given angle 7π/6 is in the third quadrant because it is greater than π and less than 3π/2.
Therefore, tan(7π/6) is negative and the angle 7π/6 lies in the third quadrant.
sorry, tan(x) is positive in QIII
Apologies for the mistake. You are correct.
To determine the sign of tan(7π/6), we need to find the value of tan(7π/6).
We know that tan(x) is positive in the third quadrant (QIII) and negative in the second quadrant (QII).
To determine the value of tan(7π/6), we can convert 7π/6 to its equivalent angle in the first revolution of the unit circle by subtracting 2π:
7π/6 - 2π = -5π/6
Now, let's find the equivalent angle with a positive value by adding 2π:
-5π/6 + 2π = 7π/6
So, tan(7π/6) has the same value as tan(7π/6) which is positive in QIII.
Therefore, tan(7π/6) is positive, and the angle 7π/6 lies in the third quadrant (QIII).
To determine the sign of tan(7π/6) and the quadrant in which it lies, we need to follow a few steps:
Step 1: Find the reference angle
The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis. In this case, the angle is 7π/6. To find the reference angle, we need to subtract the nearest multiple of π or 180 degrees. Since 7π/6 is larger than π (180 degrees) and smaller than 2π (360 degrees), we can subtract π (180 degrees) to find the reference angle.
Reference angle = 7π/6 - π = π/6
Step 2: Determine the quadrant
The quadrant in which the angle lies can be determined by looking at the sign of trigonometric functions in that quadrant. However, since we already have the reference angle, we can use that to determine the quadrant.
For the tangent function, the sign depends on the sine and cosine values of the angle in the given quadrant. In the first quadrant, both sine and cosine are positive (+/+), in the second quadrant, sine is positive and cosine is negative (+/-), in the third quadrant, both sine and cosine are negative (-/-), and in the fourth quadrant, sine is negative and cosine is positive (-/+).
Step 3: Determine the sign of tangent
Since the reference angle is π/6, we can use the unit circle or trigonometric identities to determine the sine and cosine values.
On the unit circle, π/6 corresponds to the point (√3/2, 1/2). The sine of π/6 is 1/2 and the cosine is √3/2.
Now we can determine the sign of the tangent. Tangent is defined as the ratio of sine to cosine.
tan(7π/6) = sin(π/6)/cos(π/6) = (1/2)/(√3/2) = 1/√3
Simplifying further, we rationalize the denominator by multiplying both the numerator and denominator by √3:
tan(7π/6) = (1/√3)*(√3/√3) = √3/3
So, the sign of tan(7π/6) is positive, since the numerator (√3) is positive, and the denominator (3) is positive.
The angle lies in the third quadrant because the reference angle is π/6 (30 degrees), which lies in the first quadrant, and adding π (180 degrees) to the angle places it in the third quadrant.