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Solution: Let B = A + H. Then tan(α) = A x and tan(β) = B x so α = arctan( A/x ) and β = arctan( B/x ). The viewing angle is θ = β – α = arctan( B/x ) – arctan( A/x ). We can maximize θ by calculating the derivative dθ dx and finding where the derivative is zero. Since θ = arctan( B/x ) – arctan( A/x ) , dθ
The Rule: . arctan. OR . arctan. Putting everything together: . Use this rule when you have a fraction of the form: where the polynomial in the denominator does . not factor and the fraction is not in the correct form to turn into an . Example: evaluate . From the example above we know that .
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The Mathematical Functions Site
A Taylor series for the function arctan. The integral If we invert y = arctan(x) to obtain x = tan y, then, by differentiating with respect to y, we find dx/dy = sec2 y = 1 + tan2 y = 1 + x2. Thus we have (ignoring the constant of integration) If we now differentiate. = arctan(x) = Z dx. .
Rule: If p-1∈ℤ+ ∧ c2 d2 +e2 ⩵ 0∧ (m q)∈ℤ∧ q≠ -1, let u→∫(f x)m (d+e x)q ⅆx, then f x m (d + e x) q (a + b ArcTan[c x]) p ⅆ x
The sine, cosine and tangent of an angle are all defined in terms of trigonometry, but they can also be expressed as functions. In this unit we examine these functions and their graphs. We also see how to restrict the domain of each function in order to define an inverse function.
