Answered

There are three different versions of the Pythagorean Identities. We will use $\sin^2(\theta) + \cos^2(\theta) = 1$ to obtain the other two versions.


AAssuming that $\cos(\theta)

eq 0$, divide both sides of the identity $\sin^2(\theta) + \cos^2(\theta) = 1$ by $\cos^2(\theta)$. What do we get?


BWrite the result from part (a) in the form: $(\dots)^2 + 1 = (\dots)^2$


CSimplify the expression in part (b) by recognizing the equivalent formulas for the terms inside the parentheses.


DAssuming that $\sin(\theta)

eq 0$, divide both sides of the identity $\sin^2(\theta) + \cos^2(\theta) = 1$ by $\sin^2(\theta)$. What do we get?


EWrite the result from part (d) in the form: $1 + (\dots)^2 = (\dots)^2$



(f) Simplify the expression in part (e) by recognizing the equivalent formulas for the terms inside the parentheses.



(g) Check your results in parts (c) and (f) with the formula sheet.

Verified Answer :

  • The Pythagorean identities are \(\tan^2(\theta) + 1 = \sec^2(\theta)\) and \(1 + \cot^2(\theta) = \csc^2(\theta)\).
  • Double-angle formulas: \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\) and \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\).
  • The arithmetic calculation results in \(\left(-\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \boxed{-\frac{7}{25}}\).

Explanation

  1. Analysis and Derivations

We analyze the requested mathematical tasks:

  1. Derivation of Pythagorean Identities: Starting from the fundamental identity \(\sin^2(\theta) + \cos^2(\theta) = 1\), we derive the other two identities by division.
  • Dividing by \(\cos^2(\theta)\): \(\frac{\sin^2(\theta)}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)}\). This simplifies to \(\tan^2(\theta) + 1 = \sec^2(\theta)\).
  • Dividing by \(\sin^2(\theta)\): \(\frac{\sin^2(\theta)}{\sin^2(\theta)} + \frac{\cos^2(\theta)}{\sin^2(\theta)} = \frac{1}{\sin^2(\theta)}\). This simplifies to \(1 + \cot^2(\theta) = \csc^2(\theta)\).
  1. Double-Angle Formulas:
  • For \(\cos(2\theta)\), the identity is \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\).
  • For \(\tan(2\theta)\), the identity is \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\).
  1. Arithmetic Evaluation:
  • We are asked to evaluate the expression \(\left(-\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2\).
  • Calculating the squares: \(\left(-\frac{3}{5}\right)^2 = \frac{9}{25}\) and \(\left(\frac{4}{5}\right)^2 = \frac{16}{25}\).
  • Subtracting the results: \(\frac{9}{25} - \frac{16}{25} = -\frac{7}{25}\).
  • Note that this matches the form \(\cos^2(\theta) - \sin^2(\theta) = \cos(2\theta)\), implying this calculation effectively computes \(\cos(2\theta)\) where \(\cos(\theta) = -3/5\) and \(\sin(\theta) = 4/5\).
  1. Conclusion and Final Results

The requested results can be summarized as follows:

  • The Pythagorean identities derived from \(\sin^2(\theta) + \cos^2(\theta) = 1\) are \(\boxed{\tan^2(\theta) + 1 = \sec^2(\theta)}\) and \(\boxed{1 + \cot^2(\theta) = \csc^2(\theta)}\).
  • The double-angle formulas for the specified variables are:
  • \(\cos(2\beta) = \cos^2(\beta) - \sin^2(\beta)\)
  • \(\tan(2\alpha) = \frac{2\tan(\alpha)}{1 - \tan^2(\alpha)}\)
  • \(\tan(2\beta) = \frac{2\tan(\beta)}{1 - \tan^2(\beta)}\)
  • The arithmetic evaluation \(\left(-\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2\) results in:
  • \(\frac{9}{25} - \frac{16}{25} = \boxed{-\frac{7}{25}}\).
  • Regarding the rational expression provided: \(\frac{2\tan(\theta)}{1 - \tan^2(\theta)}\) with \(\tan(\theta) = -\frac{1}{54}\) illustrates the application of the \(\tan(2\theta)\) formula. The denominator \(1 - (-\frac{1}{54})^2 = 1 - \frac{1}{2916} = \frac{2915}{2916}\). Thus, \(\tan(2\theta) = \frac{-2/\sqrt{24}}{2915/2916} = -\frac{5832}{2915\sqrt{24}}\).

Examples

In navigation, calculating the trajectory of a ship or a plane often involves trigonometric identities. For example, when determining the bearing or the change in direction relative to wind resistance, pilots use double-angle formulas to convert complex angular measurements into simpler, manageable values. Understanding these identities allows for precise course adjustments based on the sine and cosine of the angle of attack, ensuring safety and fuel efficiency during long journeys.

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