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3.11: Half Angle Formulas

Difficulty Level: At Grade Created by: CK-12
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After all of your experience with trig functions, you are feeling pretty good. You know the values of trig functions for a lot of common angles, such as \begin{align*}30^\circ, 60^\circ\end{align*}30,60 etc. And for other angles, you regularly use your calculator. Suppose someone gave you an equation like this:

\begin{align*}\cos 75^\circ\end{align*}cos75

Could you solve it without the calculator? You might notice that this is half of \begin{align*}150^\circ\end{align*}150. This might give you a hint!

When you've completed this Concept, you'll know how to solve this problem and others like it where the angle is equal to half of some other angle that you're already familiar with.

Watch This

James Sousa: Half Angle Identities

Guidance

Here we'll attempt to derive and use formulas for trig functions of angles that are half of some particular value. To do this, we'll start with the double angle formula for cosine: \begin{align*}\cos 2 \theta = 1 - 2 \sin^2 \theta\end{align*}cos2θ=12sin2θ. Set \begin{align*}\theta = \frac{\alpha}{2}\end{align*}θ=α2, so the equation above becomes \begin{align*}\cos 2 \frac{\alpha}{2} = 1 - 2 \sin^2 \frac{\alpha}{2}\end{align*}cos2α2=12sin2α2.

Solving this for \begin{align*}\sin \frac{\alpha}{2}\end{align*}sinα2, we get:

\begin{align*}\cos 2 \frac{\alpha}{2} & = 1 - 2 \sin^2 \frac{\alpha}{2} \\ \cos \alpha & = 1 - 2 \sin^2 \frac{\alpha}{2} \\ 2 \sin^2 \frac{\alpha}{2} & = 1 - \cos \alpha \\ \sin^2 \frac{\alpha}{2} & = \frac{1 - \cos \alpha}{2} \\ \sin \frac{\alpha}{2} & = \pm \sqrt{\frac{1 - \cos \alpha}{2}}\end{align*}cos2α2cosα2sin2α2sin2α2sinα2=12sin2α2=12sin2α2=1cosα=1cosα2=±1cosα2

\begin{align*}\sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}}\end{align*}sinα2=1cosα2 if \begin{align*}\frac{\alpha}{2}\end{align*}α2 is located in either the first or second quadrant.

\begin{align*}\sin \frac{\alpha}{2} = - \sqrt{\frac{1 - \cos \alpha}{2}}\end{align*}sinα2=1cosα2 if \begin{align*}\frac{\alpha}{2}\end{align*}α2 is located in the third or fourth quadrant.

This formula shows how to find the sine of half of some particular angle.

One of the other formulas that was derived for the cosine of a double angle is:

\begin{align*}\cos 2 \theta = 2 \cos^2 \theta - 1\end{align*}cos2θ=2cos2θ1. Set \begin{align*}\theta = \frac{\alpha}{2}\end{align*}θ=α2, so the equation becomes \begin{align*}\cos 2 \frac{\alpha}{2} = - 1 + 2 \cos^2 \frac{\alpha}{2}\end{align*}cos2α2=1+2cos2α2. Solving this for \begin{align*}\cos \frac{\alpha}{2}\end{align*}cosα2, we get:

\begin{align*} \cos 2 \frac{\alpha}{2} & = 2 \cos^2 \frac{\alpha}{2} - 1 \\ \cos \alpha & = 2 \cos^2 \frac{\alpha}{2} -1 \\ 2 \cos^2 \frac{\alpha}{2} & = 1 + \cos \alpha \\ \cos^2 \frac{\alpha}{2} & = \frac{1 + \cos \alpha}{2} \\ \cos \frac{\alpha}{2} & = \pm \sqrt{\frac{1 + \cos \alpha}{2}}\end{align*}cos2α2cosα2cos2α2cos2α2cosα2=2cos2α21=2cos2α21=1+cosα=1+cosα2=±1+cosα2

\begin{align*}\cos \frac{\alpha}{2} = \sqrt{\frac{1 + \cos \alpha}{2}}\end{align*}cosα2=1+cosα2 if \begin{align*}\frac{\alpha}{2}\end{align*}α2 is located in either the first or fourth quadrant.

\begin{align*}\cos \frac{\alpha}{2} = - \sqrt{\frac{1 + \cos \alpha}{2}}\end{align*}cosα2=1+cosα2 if \begin{align*}\frac{\alpha}{2}\end{align*}α2 is located in either the second or fourth quadrant.

This formula shows how to find the cosine of half of some particular angle.

Let's see some examples of these two formulas (sine and cosine of half angles) in action.

Example A

Determine the exact value of \begin{align*}\sin 15^\circ\end{align*}sin15.

Solution: Using the half angle identity, \begin{align*}\alpha = 30^\circ\end{align*}α=30, and \begin{align*}15^\circ\end{align*}15 is located in the first quadrant. Therefore, \begin{align*}\sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}}\end{align*}sinα2=1cosα2.

\begin{align*}\sin 15^\circ & = \sqrt{\frac{1 - \cos 30^\circ}{2}} \\ & = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{\frac{2 - \sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}}\end{align*}sin15=1cos302=1322=2322=234

Plugging this into a calculator, \begin{align*}\sqrt{\frac{2 - \sqrt{3}}{4}} \approx 0.2588\end{align*}2340.2588. Using the sine function on your calculator will validate that this answer is correct.

Example B

Use the half angle identity to find exact value of \begin{align*}\sin 112.5^\circ\end{align*}

Solution: since \begin{align*}\sin \frac{225^\circ}{2} = \sin 112.5^\circ\end{align*}, use the half angle formula for sine, where \begin{align*}\alpha = 225^\circ\end{align*}. In this example, the angle \begin{align*}112.5^\circ\end{align*} is a second quadrant angle, and the sin of a second quadrant angle is positive.

\begin{align*}\sin 112.5^\circ & = \sin \frac{225^\circ}{2} \\ & = \pm \sqrt{\frac{1 - \cos 225^\circ}{2}} \\ & = + \sqrt{\frac{1 - \left (- \frac{\sqrt{2}}{2} \right )}{2}} \\ & = \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} \\ & = \sqrt{\frac{2 + \sqrt{2}}{4}}\end{align*}

Example C

Use the half angle formula for the cosine function to prove that the following expression is an identity: \begin{align*}2 \cos^2 \frac{x}{2} - \cos x = 1\end{align*}

Solution: Use the formula \begin{align*}\cos \frac{\alpha}{2} = \sqrt{\frac{1 + \cos \alpha}{2}}\end{align*} and substitute it on the left-hand side of the expression.

\begin{align*}2 \left( \sqrt{\frac{1 + \cos \theta}{2}} \right )^2 - \cos \theta & = 1 \\ 2 \left (\frac{1 + \cos \theta}{2} \right ) - \cos \theta & = 1 \\ 1 + \cos \theta - \cos \theta & = 1 \\ 1 & = 1\end{align*}

Vocabulary

Half Angle Identity: A half angle identity relates the a trigonometric function of one half of an argument to a set of trigonometric functions, each containing the original argument.

Guided Practice

1. Prove the identity: \begin{align*}\tan \frac{b}{2} = \frac{\sec b}{\sec b \csc b + \csc b}\end{align*}

2. Verify the identity: \begin{align*}\cot \frac{c}{2} = \frac{\sin c}{1 - \cos c}\end{align*}

3. Prove that \begin{align*}\sin x \tan \frac{x}{2} + 2 \cos x = 2 \cos^2 \frac{x}{2}\end{align*}

Solutions:

1.

Step 1: Change right side into sine and cosine.

\begin{align*}\tan \frac{b}{2} & = \frac{\sec b}{\sec b \csc b + \csc b} \\ & = \frac{1}{\cos b} \div \csc b (\sec b + 1) \\ & = \frac{1}{\cos b} \div \frac{1}{\sin b} \left (\frac{1}{\cos b} + 1 \right ) \\ & = \frac{1}{\cos b} \div \frac{1}{\sin b} \left (\frac{1 + \cos b}{\cos b} \right ) \\ & = \frac{1}{\cos b} \div \frac{1 + \cos b}{\sin b \cos b} \\ & = \frac{1}{\cos b} \cdot \frac{\sin b \cos b}{1 + \cos b} \\ & = \frac{\sin b}{1 + \cos b}\end{align*}

Step 2: At the last step above, we have simplified the right side as much as possible, now we simplify the left side, using the half angle formula.

\begin{align*}\sqrt{\frac{1 - \cos b}{1 + \cos b}} & = \frac{\sin b}{1 + \cos b} \\ \frac{1 - \cos b}{1 + \cos b} & = \frac{\sin^2 b}{(1 + \cos b)^2} \\ (1 - \cos b)(1 + \cos b)^2 & = \sin^2 b (1 + \cos b) \\ (1 - \cos b)(1 + \cos b) & = \sin^2 b \\ 1 - \cos^2 b & = \sin^2 b\end{align*}

2.

Step 1: change cotangent to cosine over sine, then cross-multiply.

\begin{align*}\cot \frac{c}{2} & = \frac{\sin c}{1 - \cos c} \\ & = \frac{\cos \frac{c}{2}}{\sin \frac{c}{2}} = \sqrt{\frac{1 + \cos c}{1 - \cos c}} \\ \sqrt{\frac{1 + \cos c}{1 - \cos c}} & = \frac{\sin c}{1 - \cos c} \\ \frac{1 + \cos c}{1 - \cos c} & = \frac{\sin^2 c}{(1 - \cos c)^2} \\ (1 + \cos c)(1 - \cos c)^2 & = \sin^2 c (1 - \cos c) \\ (1 + \cos c)(1 - \cos c) & = \sin^2 c \\ 1 - \cos^2 c & = \sin^2 c\end{align*}

3.

\begin{align*}\sin x \tan \frac{x}{2} + 2 \cos x & = \sin x \left(\frac{1 - \cos x}{\sin x} \right ) + 2 \cos x \\ \sin x \tan \frac{x}{2} + 2 \cos x & = 1 - \cos x + 2 \cos x \\ \sin x \tan \frac{x}{2} + 2 \cos x & = 1 + \cos x \\ \sin x \tan \frac{x}{2} + 2 \cos x & = 2 \cos^2 \frac{x}{2} \end{align*}

Concept Problem Solution

The original question asked you to find \begin{align*}\cos 75^\circ\end{align*}. If you use the half angle formula, then \begin{align*}\alpha = 150^\circ\end{align*}

Substituting this into the half angle formula:

\begin{align*}\sin \frac{150^\circ}{2} = \sqrt{\frac{1 - \cos \alpha}{2}} = \sqrt{\frac{1 - \cos 150^\circ}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2}\end{align*}

Practice

Use half angle identities to find the exact values of each expression.

  1. \begin{align*}\sin 22.5^\circ\end{align*}
  2. \begin{align*}\sin 75^\circ\end{align*}
  3. \begin{align*}\sin 67.5^\circ\end{align*}
  4. \begin{align*}\sin 157.5^\circ\end{align*}
  5. \begin{align*}\cos 22.5^\circ\end{align*}
  6. \begin{align*}\cos 75^\circ\end{align*}
  7. \begin{align*}\cos 157.5^\circ\end{align*}
  8. \begin{align*}\cos 67.5^\circ\end{align*}
  9. Use the two half angle identities presented in this concept to prove that \begin{align*}\tan(\frac{x}{2})=\pm \sqrt{\frac{1-\cos x}{1+\cos x}}\end{align*}.
  10. Use the result of the previous problem to show that \begin{align*}\tan(\frac{x}{2})=\frac{1-\cos x}{\sin x}\end{align*}.
  11. Use the result of the previous problem to show that \begin{align*}\tan(\frac{x}{2})=\frac{\sin x}{1+\cos x}\end{align*}.

Use half angle identities to help you find all solutions to the following equations in the interval \begin{align*}[0,2\pi)\end{align*}.

  1. \begin{align*}\sin^2 x=\cos^2(\frac{x}{2})\end{align*}
  2. \begin{align*}\tan(\frac{x}{2})=\frac{1-\cos x}{1+\cos x}\end{align*}
  3. \begin{align*}\cos^2 x=\sin^2(\frac{x}{2})\end{align*}
  4. \begin{align*}\sin^2(\frac{x}{2})=2\cos^2 x-1\end{align*}

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Vocabulary

Half Angle Identity

A half angle identity relates a trigonometric function of one half of an argument to a set of trigonometric functions, each containing the original argument.

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Sep 26, 2012
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