If tan theta = sin theta, and theta is greater than or equal to 0 and less than or equal to 2 pi. find cos theta and what is theta
tanØ = sinØ , 0 ≤ Ø ≤ 2π
sinØ/cosØ = sinØ
sinØ = sinØcosØ
sinØ - sinØcosØ = 0
sinØ(1 - cosØ)=0
sinØ = 0 or cosØ = 1
Ø = 0, π , 2π or Ø = 0 , 2π
Ø = 0 , π , 2π
then if Ø = 0 or 2π , cosØ = 1
if Ø = π , then cosØ = -1
To find the value of cos(theta) when tan(theta) = sin(theta), we can use the trigonometric identity:
tan^2(theta) + 1 = sec^2(theta)
Since tan(theta) = sin(theta), we can substitute sin(theta) for tan(theta) in the above identity:
sin^2(theta) + 1 = sec^2(theta)
Now, let's solve for cos(theta). We know that sec(theta) is the reciprocal of cos(theta), so we can rewrite the equation as:
1 + cos^2(theta) = 1/cos^2(theta)
If we multiply both sides by cos^2(theta), we get:
cos^2(theta) + cos^4(theta) = 1
Let's simplify this equation by substituting x = cos^2(theta):
x + x^2 = 1
Rearranging the equation and setting it equal to zero, we have a quadratic equation:
x^2 + x - 1 = 0
Using the quadratic formula, we can solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 1, and c = -1:
x = (-1 ± √(1 - 4(1)(-1))) / 2(1)
x = (-1 ± √(1 + 4)) / 2
x = (-1 ± √5) / 2
Since cos(theta) is a real number, we discard the negative value:
x = (-1 + √5) / 2
Now, we take the square root of x to find cos(theta):
cos(theta) = √(((-1 + √5) / 2))
To find the value of theta, we need to find the angle whose sine and tangent are equal. One such angle is theta = π/4 (45 degrees) or 5π/4 (225 degrees) because sin(π/4) = tan(π/4) = 1.
Therefore, the value of cos(theta) is (√(((-1 + √5) / 2))) and theta can be π/4 or 5π/4.