When a pilot banks an aircraft moving at a speed, v, in metres per second, at an angle a, the radius of the turn that results is given by the formula:
r = v^2/g * tan(pi/2 - a)
where g is the accelatation due to gravity, or 9.8 m/s^2.
Use an appropriate equivalent trigonometric expression to show that this formula can be simplified to:
r= v^2 / (g*tana)
pi/2-a and a are complementary angles.
the trig ratio of any angle is the co-ratio of the complement of that angle.
e.g. sin 70 = cos 20
sec 25 = csc 65
and tan (pi/2 - a) = cot a
so
r = v^2/g * tan(pi/2 - a)
= v^2/g * cot a
= v^2/g * 1/tan a
= v^2/(g*tan a)
explain why tan 60degree=root3.
To simplify the formula r = v^2/g * tan(pi/2 - a) using an equivalent trigonometric expression, we can start by substituting the value of tan(pi/2 - a) from the trigonometric identity tan(pi/2 - a) = cot(a).
r = v^2/g * cot(a)
Next, we can rewrite cot(a) as 1/tan(a) using another trigonometric identity.
r = v^2/g * (1/tan(a))
Simplifying further, we can divide v^2 by g to move it outside the parentheses.
r = (v^2/g)*(1/tan(a))
Finally, combining the two fractions, we can rewrite the expression as follows:
r = v^2 / (g*tan(a))
Therefore, the simplified expression for the radius of the turn is r = v^2 / (g*tan(a)).
To simplify the given formula and obtain the expression r = v^2 / (g * tan(a)), we can utilize the trigonometric identity:
tan(pi/2 - a) = 1 / tan(a)
This trigonometric identity states that the tangent of the complementary angle is equal to the reciprocal of the tangent of the original angle.
Now, let's substitute this equivalent trigonometric expression into the original formula and simplify:
r = v^2/g * tan(pi/2 - a)
r = v^2/g * (1 / tan(a))
r = v^2 / (g * tan(a))
Hence, by applying the trigonometric identity tan(pi/2 - a) = 1 / tan(a), we simplified the given formula to r = v^2 / (g * tan(a)).