Find the exact values.
(a) tan(π/3) = ?
(b) sin(7π/6) = ?
(c) sec(5π/3)= ?
To find the exact values of trigonometric functions, we need to use the unit circle or reference angles.
(a) tan(π/3) = ?
To find the exact value of tan(π/3), we can use the unit circle. First, we need to determine the reference angle for π/3. A reference angle is the acute angle formed between the terminal side of the angle and the x-axis.
In this case, π/3 is in the first quadrant of the unit circle, so the reference angle is also π/3.
Next, we need to determine the values of sin(π/3) and cos(π/3). For π/3, sin(π/3) = √3/2 and cos(π/3) = 1/2.
Finally, we can use the formula: tan(theta) = sin(theta) / cos(theta).
So, at π/3, tan(π/3) = sin(π/3) / cos(π/3) = (√3/2) / (1/2) = √3.
Therefore, tan(π/3) = √3.
(b) sin(7π/6) = ?
To find the exact value of sin(7π/6), we again need to use the unit circle. First, let's determine the reference angle for 7π/6.
Since 7π/6 is in the third quadrant of the unit circle, the reference angle is π/6.
Next, we determine sin(π/6) and cos(π/6). For π/6, sin(π/6) = 1/2 and cos(π/6) = √3/2.
However, since 7π/6 is in the third quadrant, both sin and cos will be negative.
So, sin(7π/6) = -sin(π/6) = -(1/2) = -1/2.
Therefore, sin(7π/6) = -1/2.
(c) sec(5π/3) = ?
To find the exact value of sec(5π/3), we can use the reciprocal relationship between sec and cos.
First, we determine cos(5π/3) by using the unit circle. For 5π/3, cos(5π/3) = -1/2.
Then, we use the formula: sec(theta) = 1 / cos(theta).
So, sec(5π/3) = 1 / cos(5π/3) = 1 / (-1/2) = -2.
Therefore, sec(5π/3) = -2.
You need to review your commonly used reference angles.
sin π/6 = 1/2
sin π/3 = √3/2
Now you should have no trouble.