Evaluate the sine, cosine, tangent, cosecant, secant, and cotangent of t= − 2π/3

Question

If the trigonometric function is undefined, enter "u".

sin(− 2π/3)= ?

cos(− 2π/3)= ?

tan(− 2π/3)= ?

csc(− 2π/3)= ?

sec(− 2π/3)= ?

cot(− 2π/3)= ?

Answer 1

pi=180 -2(180)/3=-120 and -120 corresponds to 60 and since the radian is negative -120 will be in the third quadrant. since -2pi/3 is equal to -120 sin(-120) is similar to sin(60) and sin(60)=sqrt(3)/2 so then sin(-120)=-sqrt(3)/2. Cos(-120) is also similar to cos(60) due to sin and cos sharing the same angle which also resembles a right triangle where Pythagorean Theorem is used. So cos(60)=sqrt(1)/2 or 1/2 so then cos(-120)=-1/3. Tan(-120) is the ratio between sin and cos such that sin/cos=tan so then Tan(-120)=-sqrt(3)/2 divided by -1/3 and by using keep change flip rule you get -sqrt(3)/2 times -2/1 equals sqrt(3) since 2/2 is 1 and -sqrt(3) divided by -1 equals sqrt(3). Csc is the inverse of sin so Csc(-120)=-2/sqrt(3) and rationalize the denominator to get -2sqrt(3)/3. Sec(-120) is the inverse of Cos(-120) so Sec(-120)=-2/1 or just -2. Cot(-120) is also the inverse of tan(-120) so then Cot(-120)=1/sqrt(3) and then rationalize the denomiantor so it then becomes sqrt(3)/3.