Please find the exact values:
Tan(-3p/4)=
Sec(2p/3)=
Csc(7p/4)=
Cot(11p/6)=
p=pie
-3pi/4 = 2pi -3pi/4 = + 5 pi/4
so you really want tan 5pi/4 which is pi/4 below the -y axis
x = -1
y = -1
y/x = tangent = 1
sec 2 pi/3 = 1/cos 2 pi/3
that angle is between pi/2 and pi, second quadrant
pi - 2 pi/3 = pi/3 above -y axis
x = -1
y = +sqrt 3
sqrt(x^2+y^2) = hypotenuse = 2
so
cos(2pi/3) = -1/2
sec(2 pi/3)= -2
etc. draw the triangles
You should be familiar with the trig ratios of the 30-60-90º triangle as well as the 45-45-90º triangle
pi/4 radians <-----> 45º
pi/3 radians <-----> 60º
pi/6 radians <-----> 30º
all your given angles are multiples of these.
I will do the last one
cot(11pi/6) = 1/tan(11pi/6)
11pi/6 is in the fourth quadrant, in the fourth quadrant the tangent is negative, so the cotangent is negative.
The angle is pi/6 away from the x-axis, so the 'angle in standard position' is pi/6
tan pi/6 = 1/√3, so cot pi/6 = √3
then cot(11pi/6) = -√3
do the others the same way.
What is sin of 660?
And
What is csc(-2pi)/3?
To find the exact values of these trigonometric functions, we will make use of the unit circle and the reference angle for each given angle.
1. Tan(-3π/4):
Since the angle is negative, it lies in the third quadrant on the unit circle. The reference angle for -3π/4 is π/4.
In the third quadrant, tangent is negative, so tan(-3π/4) = -tan(π/4).
The value of tangent for π/4 is 1, so tan(-3π/4) = -1.
2. Sec(2π/3):
The angle 2π/3 lies in the second quadrant on the unit circle. To find secant, we need to find the reciprocal of the cosine.
The reference angle for 2π/3 is π/3.
The value of cosine for π/3 is 1/2, so sec(2π/3) = 1/(cos(π/3)) = 1/(1/2) = 2.
3. Csc(7π/4):
The angle 7π/4 lies in the fourth quadrant on the unit circle. To find cosecant, we need to find the reciprocal of the sine.
The reference angle for 7π/4 is π/4.
The value of sine for π/4 is 1/√2, so csc(7π/4) = 1/(sin(π/4)) = 1/(1/√2) = √2.
4. Cot(11π/6):
The angle 11π/6 lies in the fourth quadrant on the unit circle. To find cotangent, we need to find the reciprocal of the tangent.
The reference angle for 11π/6 is π/6.
The value of tangent for π/6 is √3, so cot(11π/6) = 1/(tan(π/6)) = 1/√3.
In summary:
Tan(-3π/4) = -1
Sec(2π/3) = 2
Csc(7π/4) = √2
Cot(11π/6) = 1/√3