To find the values of cos and tan given sin 1/2 and sin 3/2, we can use the Pythagorean identity and the definitions of the trigonometric functions.
First, let’s clarify what sin 1/2 and sin 3/2 mean. In the context of the unit circle, we have:
- sin(1/2) is the sine of the angle π/6 radians (or 30 degrees), which is equal to 0.5.
- sin(3/2) is likely a reference to the angle 3π/2 radians (or 270 degrees), where the sine value is -1.
Next, we apply the Pythagorean identity:
sin²(x) + cos²(x) = 1
For the first case, when sin = 1/2:
- sin²(1/2) = (1/2)² = 1/4
- Thus, cos²(1/2) = 1 – 1/4 = 3/4
- Therefore, cos(1/2) = ±√(3/4) = ±√3/2.
For tangent:
tan(x) = sin(x) / cos(x)
So, tan(1/2) = (1/2) / (√3/2) = 1/√3 = √3/3.
Now for the second case with sin = -1 (corresponding to sin(3/2)):
- Using the identity again:
- sin²(3/2) = (-1)² = 1
- Thus, cos²(3/2) = 1 – 1 = 0.
- Therefore, cos(3/2) = 0.
For tangent when sin = -1 and cos = 0:
tan(3/2) = sin(3/2) / cos(3/2) is undefined since we cannot divide by zero.
In summary:
- For sin(1/2): cos = ±√3/2, tan = √3/3.
- For sin(3/2): cos = 0, tan is undefined.