How do you find all six trigonometric functions of 240 degrees?

Question

Charles Autterson · Accepted Answer

#240^o# has a reference angle of #60^o# as indicated in the image below. A #60^o# angle is a basic angle from one of the common triangles:

From their definitions:

#sin(240^o) = -sqrt(3)/2#

#cos(240^o) = -1/2#

#tan(240^o) = sqrt(3)#

#csc(240^o) = - 2/sqrt(3)#

#sec(240^o) = -2#

#cot(240^o) = 1/sqrt(3)#

Elijah Alexander · Answer

There is another way, using the trig unit circle. #sin 240 = sin (60 + 180) = -sin 60 = -(sqr3)/2# (trig table) #cos 240 = cos (60 + 180) = -cos 60 = -1/2# #tan 240 = sin 240/cos 240 = sqr3# #cot 240 = = 1/(sqr3) = (sqr3)/3# #sec 240 = 1/sin 240 = - (2sqr3)/3# #csc 240 = 1/cos 240 = -2#

Quinn Anderson · Answer

undefined To find all six trigonometric functions of 240 degrees, follow these steps:

1. Convert 240 degrees to radians. Recall that 1 degree is equal to π/180 radians.

$$ 240^\circ \times \frac{\pi}{180} = \frac{4\pi}{3} $$

2. Identify the reference angle. Since 240 degrees is in the third quadrant, its reference angle in the first quadrant is 180 degrees - 240 degrees = 60 degrees.

3. Determine the sign of each trigonometric function based on the quadrant:
   - In the third quadrant, sine and cosecant are negative, while cosine, secant, tangent, and cotangent are positive.

4. Calculate the trigonometric functions using the reference angle and the signs determined in step 3:
   - Sine (sin): $$ \sin(240^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2} $$
   - Cosine (cos): $$ \cos(240^\circ) = -\cos(60^\circ) = -\frac{1}{2} $$
   - Tangent (tan): $$ \tan(240^\circ) = -\tan(60^\circ) = -\sqrt{3} $$
   - Cosecant (csc): $$ \csc(240^\circ) = -\csc(60^\circ) = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} $$
   - Secant (sec): $$ \sec(240^\circ) = -\sec(60^\circ) = -\frac{2}{1} = -2 $$
   - Cotangent (cot): $$ \cot(240^\circ) = -\cot(60^\circ) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $$

These are the values of the six trigonometric functions for the angle 240 degrees.