Evaluate each of the following. Leave results as exact values.
(a) tan(5π/6) =
(b) tan(4π/3) =
(c) cot(7π/4) =
(d) sec(2π/3) =
(e) csc(3π/4) =
My answers
(a) tan(5π/6) = -.577350
(b) tan(4π/3) = 1.732051
(c) cot(7π/4) = -1
(d) sec(2π/3) = -2
(e) csc(3π/4) = 1.414214
That's not what they want.
When it says "exact" values, you need answers that are either
rational numbers of results expressed as square roots, etc
These kind of questions usually involve standard angles, that is,
angles whose trig ratios you should know
You must also know the patterns of the CAST rule.
Some people have an easier time thinking in terms of degrees vs radians
e.g. (a) tan(5π/6) = ??
is the same as tan(150°)
and 150° has 30° as its angle in standard position
You must know tan 30° = 1/√3
and since 150° is in quadrant II, and by the CAST rule the tangent is
negative in quad II
tan 150° = -1/√3
tan(5π/6) = -1/√3 or -√3 / 3 if you need it rationalized.
one more:
csc(3π/4)
= csc 135° , in quad II where the sine and cosecant are positive
= + csc 45°
= 1/sin 45° = 1/(1/√2)
= √2
They do want it in exact form though
I submited it and 40% was right
They should not have accepted those answers, except c) and d)
or else drop the "exact" answer request.
To evaluate each of the trigonometric functions, we can use the unit circle and the key values for sine (sin) and cosine (cos) of common angles.
(a) To evaluate tan(5π/6), we need to find the sine and cosine of 5π/6.
The cosine of 5π/6 can be found by looking at the unit circle. In the second quadrant, the x-coordinate (cosine) is negative, so cos(5π/6) = -1/2.
The sine of 5π/6 can also be found using the unit circle. In the second quadrant, the y-coordinate (sine) is positive, so sin(5π/6) = √3/2.
Now we can evaluate tan(5π/6) as tan(5π/6) = sin(5π/6) / cos(5π/6) = (√3/2) / (-1/2) = -√3.
So, tan(5π/6) = -√3.
(b) To evaluate tan(4π/3), we need to find the sine and cosine of 4π/3.
The cosine of 4π/3 can be found by looking at the unit circle. In the third quadrant, the x-coordinate (cosine) is negative, so cos(4π/3) = -1/2.
The sine of 4π/3 can also be found using the unit circle. In the third quadrant, the y-coordinate (sine) is negative, so sin(4π/3) = -√3/2.
Now we can evaluate tan(4π/3) as tan(4π/3) = sin(4π/3) / cos(4π/3) = (-√3/2) / (-1/2) = √3.
So, tan(4π/3) = √3.
(c) To evaluate cot(7π/4), we need to find the sine and cosine of 7π/4.
The cosine of 7π/4 can be found by looking at the unit circle. In the fourth quadrant, the x-coordinate (cosine) is positive, so cos(7π/4) = 1/√2 = √2/2.
The sine of 7π/4 can also be found using the unit circle. In the fourth quadrant, the y-coordinate (sine) is negative, so sin(7π/4) = -1/√2 = -√2/2.
Now we can evaluate cot(7π/4) as cot(7π/4) = cos(7π/4) / sin(7π/4) = (√2/2) / (-√2/2) = -1.
So, cot(7π/4) = -1.
(d) To evaluate sec(2π/3), we need to find the cosine of 2π/3.
The cosine of 2π/3 can be found by looking at the unit circle. In the second quadrant, the x-coordinate (cosine) is negative, so cos(2π/3) = -1/2.
Now we can evaluate sec(2π/3) as sec(2π/3) = 1/cos(2π/3) = 1 / (-1/2) = -2.
So, sec(2π/3) = -2.
(e) To evaluate csc(3π/4), we need to find the sine of 3π/4.
The sine of 3π/4 can be found by looking at the unit circle. In the second quadrant, the y-coordinate (sine) is positive, so sin(3π/4) = √2/2.
Now we can evaluate csc(3π/4) as csc(3π/4) = 1/sin(3π/4) = 1 / (√2/2) = √2.
So, csc(3π/4) = √2.
Therefore, the correct answers are:
(a) tan(5π/6) = -√3
(b) tan(4π/3) = √3
(c) cot(7π/4) = -1
(d) sec(2π/3) = -2
(e) csc(3π/4) = √2.