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Finding Exact Trigonometric Values Using Sum and Difference Formulas

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Finding Exact Trig Values using Sum and Difference Formulas

You measure an angle with your protractor to be 165^\circ . How could you find the exact sine of this angle without using a calculator?

Guidance

You know that \sin 30^\circ=\frac{1}{2}, \cos 135^\circ=-\frac{\sqrt{2}}{2}, \tan 300 ^\circ = -\sqrt{3}, etc... from the special right triangles. In this concept, we will learn how to find the exact values of the trig functions for angles other than these multiples of 30^\circ, 45^\circ, and 60^\circ . Using the Sum and Difference Formulas, we can find these exact trig values.

Sum and Difference Formulas

\sin(a\pm b) &=\sin a \cos b \pm \cos a \sin b \\\cos(a\pm b) &=\cos a \cos b \pm \sin a \sin b \\\tan(a \pm b) &=\frac{\tan a \pm \tan b}{1 \pm \tan a \tan b}

Example A

Find the exact value of \sin 75^\circ .

Solution: This is an example of where we can use the sine sum formula from above, \sin(a+b)=\sin a \cos b+\cos a \sin b , where a = 45^\circ and b = 30^\circ .

\sin 75^\circ &=\sin(45^\circ + 30 ^\circ) \\&= \sin 45^\circ \cos 30^\circ +\cos 45^\circ \sin 30 ^\circ \\&= \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2} \\&= \frac{\sqrt{6}+\sqrt{2}}{4}

In general, \sin (a+b)\ne \sin a+\sin b and similar statements can be made for the other sum and difference formulas.

Example B

Find the exact value of \cos \frac{11 \pi}{12} .

Solution: For this example, we could use either the sum or difference cosine formula, \frac{11\pi}{12}=\frac{2\pi}{3}+\frac{\pi}{4} or \frac{11\pi}{12}=\frac{7\pi}{6}-\frac{\pi}{4} . Let’s use the sum formula.

\cos \frac{11\pi}{12} &=\cos \left(\frac{2\pi}{3}+\frac{\pi}{4}\right) \\&=\cos \frac{2\pi}{3}\cos \frac{\pi}{4}-\sin\frac{2\pi}{3}\sin \frac{\pi}{4} \\&= -\frac{1}{2}\cdot \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \\&= -\frac{\sqrt{2}+\sqrt{6}}{4}

Example C

Find the exact value of \tan \left(-\frac{\pi}{12}\right) .

Solution: This angle is the difference between \frac{\pi}{4} and \frac{\pi}{3} .

\tan \left(\frac{\pi}{4}-\frac{\pi}{3}\right) &=\frac{\tan \frac{\pi}{4}-\tan \frac{\pi}{3}}{1+\tan \frac{\pi}{4} \tan \frac{\pi}{3}} \\&=\frac{1-\sqrt{3}}{1+\sqrt{3}}

This angle is also the same as \frac{23 \pi}{12} . You could have also used this value and done \tan\left(\frac{\pi}{4}+\frac{5 \pi}{3}\right) and arrived at the same answer.

Concept Problem Revisit

We can use the sine sum formula, \sin(a+b)=\sin a \cos b+\cos a \sin b , where a = 120^\circ and b = 45^\circ .

\sin 165^\circ &=\sin(120^\circ + 45 ^\circ) \\&= \sin 120^\circ \cos 45^\circ +\cos 120^\circ \sin 45 ^\circ \\&= \frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2}+\frac{-1}{2} \cdot \frac{\sqrt{2}}{2} \\&= \frac{\sqrt{6}-\sqrt{2}}{4}\\

Guided Practice

Find the exact values of:

1. \cos 15^\circ

2. \tan 255^\circ

Answers

1. \cos 15^\circ &=\cos(45^\circ - 30^\circ) \\&= \cos 45^\circ \cos 30^\circ - \sin 45 ^\circ \sin 30 ^\circ \\&= \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\cdot \frac{1}{2} \\&= -\frac{\sqrt{6}-\sqrt{2}}{4}

2. \tan (210^\circ + 45^\circ) &=\frac{\tan 210^\circ+\tan 45^\circ}{1-\tan 210^\circ \tan 45^\circ} \\&= \frac{\frac{\sqrt{3}}{3}+1}{1-\frac{\sqrt{3}}{3}}=\frac{\frac{\sqrt{3}+3}{3}}{\frac{3-\sqrt{3}}{3}}=\frac{\sqrt{3}+3}{3-\sqrt{3}}

Vocabulary

Sum and Difference Formulas
\sin(a\pm b) &=\sin a \cos b \pm \cos a \sin b \\\cos (a \pm b) &=\cos a \cos b \mp \sin a \sin b \\\tan (a \pm b) &=\frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}

Practice

Find the exact value of the following trig functions.

  1. \sin 15^\circ
  2. \cos \frac{5\pi}{12}
  3. \tan 345^\circ
  4. \cos (-255^\circ)
  5. \sin \frac{13 \pi}{12}
  6. \sin \frac{17\pi}{12}
  7. \cos 15^\circ
  8. \tan (-15^\circ)
  9. \sin 345^\circ
  10. Now, use \sin 15^\circ from #1, and find \sin 345^\circ . Do you arrive at the same answer? Why or why not?
  11. Using \cos 15^\circ from #7, find \cos 165^\circ . What is another way you could find \cos 165^\circ ?
  12. Describe any patterns you see between the sine, cosine, and tangent of these “new” angles.
  13. Using your calculator, find the \sin 142^\circ . Now, use the sum formula and your calculator to find the \sin 142^\circ using 83^\circ and 59^\circ .
  14. Use the sine difference formula to find \sin 142^\circ with any two angles you choose. Do you arrive at the same answer? Why or why not?
  15. Challenge Using \sin (a+b)=\sin a \cos b +\cos a \sin b and \cos (a+b)=\cos a \cos b - \sin a \sin b , show that \tan (a+b)=\frac{\tan a + \tan b}{1-\tan a \tan b} .

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