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Simplifying Trigonometric Expressions using Sum and Difference Formulas

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Simplifying Trig Expressions using Sum and Difference Formulas

As Agent Trigonometry you are given this clue: \sin(\frac{\pi}{2} - x) . How could you simplify this expression to make solving your case easier?

Guidance

We can also use the sum and difference formulas to simplify trigonometric expressions.

Example A

The \sin a = -\frac{3}{5} and \cos b =\frac{12}{13} . a is in the 3^{rd} quadrant and b is in the 1^{st} . Find \sin(a+b) .

Solution: First, we need to find \cos a and \sin b . Using the Pythagorean Theorem, missing lengths are 4 and 5, respectively. So, \cos a=-\frac{4}{5} because it is in the 3^{rd} quadrant and \sin b = \frac{5}{13} . Now, use the appropriate formulas.

\sin (a+b) &=\sin a \cos b + \cos a \sin b \\&= -\frac{3}{5}\cdot \frac{12}{13}+-\frac{4}{5}\cdot \frac{5}{13} \\&= -\frac{56}{65}

Example B

Using the information from Example A, find \tan (a+b) .

Solution: From the cosine and sine of a and b , we know that \tan a=\frac{3}{4} and \tan b=\frac{5}{12} .

\tan (a+b) &=\frac{\tan a +\tan b}{1-\tan a \tan b} \\&= \frac{\frac{3}{4}+\frac{5}{12}}{1-\frac{3}{4}\cdot\frac{5}{12}} \\&= \frac{\frac{14}{12}}{\frac{11}{16}}=\frac{56}{33}

Example C

Simplify \cos (\pi - x) .

Solution: Expand this using the difference formula and then simplify.

\cos (\pi - x) &=\cos \pi \cos x +\sin \pi \sin x \\&=-1\cdot \cos x +0\cdot \sin x \\&=-\cos x

Concept Problem Revisit You can expand the expression using the difference formula and then simplify.

\sin(\frac{\pi}{2} - x)=\sin \frac{\pi}{2} \cos x - \cos \frac{\pi}{2} \sin x \\&=1\cdot \cos x - 0\cdot \sin x \\&=cos x

Guided Practice

1. Using the information from Example A, find \cos(a-b) .

2. Simplify \tan (x+\pi) .

Answers

1. \cos(a-b) &=\cos a \cos b + \sin a \sin b \\&=-\frac{4}{5}\cdot \frac{12}{13}+-\frac{3}{5}\cdot\frac{5}{13} \\&=-\frac{63}{65}

2. \tan (x+\pi)&=\frac{\tan x +\tan \pi}{1-\tan x \tan \pi} \\&=\frac{\tan x +0}{1-\tan 0} \\&=\tan x

Practice

\sin a =-\frac{8}{17}, \pi \le a < \frac{3\pi}{2} and \sin b =-\frac{1}{2}, \frac{3\pi}{2}\le b <2\pi . Find the exact trig values of:

  1. \sin (a+b)
  2. \cos (a+b)
  3. \sin (a-b)
  4. \tan (a+b)
  5. \cos (a-b)
  6. \tan (a-b)

Simplify the following expressions.

  1. \sin (2\pi-x)
  2. \sin \left(\frac{\pi}{2}+x\right)
  3. \cos (x+\pi)
  4. \cos \left(\frac{3\pi}{2}-x\right)
  5. \tan(x+2\pi)
  6. \tan(x-\pi)
  7. \sin \left(\frac{\pi}{6}-x\right)
  8. \tan \left(\frac{\pi}{4}+x\right)
  9. \cos \left(x-\frac{\pi}{3}\right)

Determine if the following trig statements are true or false.

  1. \sin(\pi - x)=\sin (x-\pi)
  2. \cos(\pi - x)=\cos (x-\pi)
  3. \tan(\pi - x)=\tan (x-\pi)

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