Question

Вычислить arccos(cos(240))+arctg(tg(-120))+arcsin(sin(120))

Answer 1

Arccos(cosx)=x  , только если  0°≤х≤180° .

Так как угол в 240° не входит в указанный промежуток, то необходимо привести этот угол к промежутку [0°,180°] c помощью тригонометрических формул, учитывая периодичность и чётность тригонометрических функций.

сos240°=cos(360°-120°)=cos(-120°)=cos120° ,  120°∈[ 0°,180°]  °⇒
arccos(cos240°)=arccos(cos120°)=120°.

arctg(tgx)=x , только если  -90°<x<90° .

tg(-120°)= -tg(120°)= -tg(180°-60°)=-(-tg60°)=tg60°  ,  60°∈(-90°,90°)  ⇒
arctg(tg(-120°))=arctg(tg60°)=60°

arcsin(sinx)=x , только если  -90°≤x≤90° .

sin120°=sin(180°-60°)=sin60°  ,  60°∈[-90°90°]  ⇒
arcsin(sin120°)=arcsin(sin60°)=60°

arccos(cos240°)+arctg(tg(-120°))+arcsin120°=120°+60°+60°=240°

Answer 2

To compute the given expression, we need to be mindful of the range of the trigonometric functions and the principal values of the inverse trigonometric functions. The principal values of inverse trigonometric functions are typically restricted to certain ranges to ensure uniqueness.

Let's calculate the expression step by step:

1. arccos(cos(240)): The angle 240 degrees lies in the third quadrant, where the cosine function is negative. The principal range for arccos is [0, π]. Since the cosine function is periodic with a period of 360 degrees (or 2π radians), we can find an equivalent angle in the principal range:

240 degrees = 240 - 360 = -120 degrees = -120 * π/180 radians

arccos(cos(-120)) = arccos(cos(-120 * π/180)) = arccos(cos(-2π/3)) = arccos(-1/2) = π/3

2. arctg(tg(-120)): The tangent function is periodic with a period of 180 degrees (or π radians). So, we can find an equivalent angle within the range of arctan, which is (-π/2, π/2):

-120 degrees = -120 + 180 = 60 degrees = 60 * π/180 radians

arctg(tg(60)) = arctg(tg(60 * π/180)) = arctg(tg(π/3)) = π/3

3. arcsin(sin(120)): The sine function is positive in the first and second quadrants. Since 120 degrees lies in the second quadrant, we need to find the equivalent angle in the principal range of arcsin, which is [-π/2, π/2]:

120 degrees = 120 - 180 = -60 degrees = -60 * π/180 radians

arcsin(sin(-60)) = arcsin(sin(-60 * π/180)) = arcsin(sin(-π/3)) = -π/3

Now, we add up the results from step 1, 2, and 3:

π/3 + π/3 - π/3 = π/3

Therefore, the value of the expression is π/3 (approximately 1.0472 radians).