Solve the equations below exactly Give your answers in radians, and find all possible values for t If there is more than one answer, enter your solutions in a comma separated list
(a) sin(t)= sqroot(2)/2 when t=
(b) cos(t)=1/2 when t=
(c) tan(t)=-1 when t=
Review your standard angles:
0, π/6, π/4, π/3, π/2
and signs in the quadrants
Then solving these will be a cinch.
To solve the equations exactly and find all possible values for t in radians, we'll use the inverse trigonometric functions. These functions help us determine the angle for which a given trigonometric ratio holds true.
(a) Given sin(t) = sqrt(2)/2, we can use the inverse of the sine function, denoted as arcsin or sin^(-1), to find the angle in radians.
arcsin(sqrt(2)/2) = π/4
So, one possible value for t is t = π/4 radians. However, keep in mind that sine is periodic with a period of 2π. Therefore, any angle that is π/4 plus an integer multiple of 2π will have the same sine ratio.
So, all possible values for t are t = π/4 + k(2π), where k is an integer.
(b) Given cos(t) = 1/2, we can use the inverse of the cosine function, denoted as arccos or cos^(-1), to find the angle in radians.
arccos(1/2) = π/3
So, one possible value for t is t = π/3 radians. Similar to sine, cosine is also periodic with a period of 2π. Therefore, any angle that is π/3 plus an integer multiple of 2π will have the same cosine ratio.
So, all possible values for t are t = π/3 + k(2π), where k is an integer.
(c) Given tan(t) = -1, we can use the inverse of the tangent function, denoted as arctan or tan^(-1), to find the angle in radians.
arctan(-1) = -π/4
So, one possible value for t is t = -π/4 radians. Tangent is also periodic, with a period of π. Therefore, any angle that is -π/4 plus an integer multiple of π will have the same tangent ratio.
So, all possible values for t are t = -π/4 + kπ, where k is an integer.