Double and Half Angle Formulas

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Textbook Reference: Section 7.3 Double-Angle, Half-Angle, and Reduction Formulas

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What's the coolest thing about circular function identities?

The fact that you can always "play" with them and discover new relationships.

Here are the identities you're going to discover next:

Double Angle Formulas

\[ \begin{align*} \sin(2\theta) &= 2\sin(\theta)\cos(\theta)\\[3 ex] \cos(2\theta) &= \cos^2(\theta)-\sin^2(\theta)\\ &= 1-2\sin^2(\theta)\\ &= 2\cos^2(\theta)-1\\[3 ex] \tan(2\theta) &= \dfrac{2\tan(\theta)}{1-\tan^2(\theta)} \end{align*} \]

Half Angle Formulas

\[ \begin{align*} \left|\sin\left(\dfrac\theta2\right)\right| &= \sqrt{\dfrac{1-\cos(\theta)}2}\\[3 ex] \left|\cos\left(\dfrac\theta2\right)\right| &= \sqrt{\dfrac{1+\cos(\theta)}2}\\[3 ex] \tan\left(\dfrac\theta2\right) &= \dfrac{1-\cos(\theta)}{\sin(\theta)}\\ &= \dfrac{\sin(\theta)}{1+\cos(\theta)} \end{align*} \] There's really not a lot to explain here — these are the formulas, and you can use them to solve similar problems to the ones you do with sum and difference identities.

The real meat of the math is in deriving them...

...which is what you'll do now!

Preview Activity 16

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

1. Let \(\theta=60^\circ\). Show how to calculate \(\sin(2\theta)\), \(\cos(2\theta)\), and \(\tan(2\theta)\) using the double-angle formulas. Your answers should end up matching the unit circle values for \(\sin(120^\circ)\), \(\cos(120^\circ)\), and \(\tan(120^\circ)\).
2. Prove the formula \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\) by starting with the sum formula \[\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)\] and letting both \(\alpha\) and \(\beta\) equal the same angle \(\theta\). (This doesn't take more than maybe two lines.)
   Prove the formulas \(\cos(2\theta) = \cos^2(\theta)-\sin^2(\theta)\) and \(\tan(2\theta) = \dfrac{2\tan(\theta)}{1-\tan^2(\theta)}\) a similar way.
3. Notice that there are three different formulas for \(\cos(2\theta)\). Show how to turn the first one into the other two using Pythagorean identities.
4. Starting with the formula \(\cos(2\theta)=1-2\sin^2(\theta)\), prove the first half-angle formula \[\left|\sin\left(\dfrac\theta2\right)\right| = \sqrt{\dfrac{1-\cos(\theta)}2}\] by replacing each \(\theta\) with \(\dfrac\theta2\) and then solving for \(\sin\left(\dfrac\theta2\right)\). Then use a similar technique to prove the cosine half-angle formula. Where do the absolute values come from?
5. (CHALLENGE — OPTIONAL!)
   Prove the tangent half-angle formulas. You'll need to use the quotient identities and introduce a cleverly-chosen factor of \(1\) into the fraction.
6. Answer AT LEAST one of the following questions:
   1. What was something you found interesting about this activity?
   2. What was an "a-ha" moment you had while doing this activity?
   3. What was the muddiest point of this activity for you?
   4. What question(s) do you have about anything you've done?

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Posted by Bill Shillito