solve for theta
6. tan theta=-1
7. sin(theta/2)=1/2
thank you for your help!
tanØ = -1
angle in standard position is 45° (tan^-1 (1) = 45° )
the tangent is negative in quads II and IV
so Ø = 180-45 = 135
or
Ø = 360-45 = 315°
check: tan135 = -1 and tan315 = -1
sin Ø/2 = 1/2
angle in standard postion is 30° and the sine is positive in I and II
Ø/2 = 30° or Ø/2 = 150°
Ø = 60° or Ø = 300°
check
sin (60/2) = 1/2 and sin (300/2) = 1/2
thank you so much! :)
To solve for theta in each of these equations, you can use trigonometric identities and properties to isolate theta.
1. For the equation 6 * tan(theta) = -1:
- Start by dividing both sides of the equation by 6: tan(theta) = -1/6.
- Next, take the inverse tangent (arctan) of both sides to eliminate the tangent function and find the angle. So, theta = arctan(-1/6).
2. For the equation sin(theta/2) = 1/2:
- Multiply both sides of the equation by 2 to get rid of the fraction: 2 * sin(theta/2) = 1.
- Since the angle is divided by 2, we can use the half-angle identity for sine: sin(theta/2) = sqrt((1 - cos(theta)) / 2).
- Substitute the given equation: sqrt((1 - cos(theta)) / 2) = 1.
- Square both sides of the equation to eliminate the square root: ((1 - cos(theta)) / 2) = 1.
- Multiply both sides by 2: 1 - cos(theta) = 2.
- Solve for cos(theta): cos(theta) = 1 - 2 = -1.
- To find theta, take the inverse cosine (arccos) of both sides: theta = arccos(-1).
Remember to use the appropriate range of angles depending on the context of the problem (e.g., degrees or radians).